With ant's there are at least 2 major obstacles, preventing them to be sapient.
- Speed of information transfer and mobility.
Also maybe more fundamental, system as whole(or some aspects of it) which makes anthill an anthill and limits size.
For ants, we can imagine system which allows to transfer and work with information, bits of it. And that system makes some useful results. We can, because they already have such systems.
But what makes system, which others have, mammals and others, useful?
- It's fast. Someone may be dumb but fast, and be successful.
- It's energy efficient comparing to what it does (helps find new energy and in some cases prevents being energy for someone else). Smarter someone is - better it might be at finding food or preventing being food himself.
- It allows to learn, tune(learn) actions fast, on the fly in some cases. Even for a snail, which needs 20-30 lessons in simple cases, which it may comprehend according to it's smartness level, it may take just a hour (not sure about time, just assume it repeats some actions once in 2 minutes, maybe a bit to fast for it, but in general 2-3 hours it think is reasonable amount of time for it, and it may be smarter then that, Helix pomatia used to be laboratory snail, because of relatively simple structure of neuron networks and bigger size of nerve cells).
Smart-fast-energy efficient - at some extend any from this triad may compensate imperfection of other two. You may be fasts, but not smart not energy efficient - as example Mustelidae in general or Polecat, Mink(just funny, mink farm) as example. Can't find atm and can not recall - saw somewho from Mustelidae smaller, just little bigger then rat, they are so crazy, so target driven, all time in movement, all brains in killing it's pray, nonstop - but also they like to play and are funny.
For wide range of combinations, it can be found a species, which will express this combination the most, air, ground, see, deep see. But brain principles(I mean neuron cells) are same. It shows how flexible that system is, it works from no brainers to relatively smart creatures.
In all these cases speed of signal (20 m/s, differs in some range for different species), efficiency(element is few cells, even if the are bigger then usually still a cell, so they consume less energy then significant size multicellular organism), compactness (it's a boost for and speed, and efficiency, and mobility) - these properties are vital for systems, which we may observe atm.
For system based on ants interaction, as separate elements of such system
Can it be fast, as information processing? With chemical and physical approach, no. Way to slow, just reacting to food stimulus in their current stage it may take them 15-30 minutes to react on food presence. And it's kinda typical time of their thinking, few bits tick.
Ants as messengers, as information carriers, they have zipp back and forth at 20m/s speeds and actually way more then that - depends on size of that ants-brain - to gain proper speed of thinking.
Dense packing of information and processing it - when it goes about mm size creatures. If we take big cell 0.1mm and small ant 1mm - there difference in volume 1000 times, roughly speaking 1000 times difference in energy consumption for 1 bit processing, storing. If their speed is 2 cm/s it means 1000 times slower signal processing, and because of size which will be 10 times bigger (linear difference instead of volume) it will be not 1000 but 10000 times slower.
So to reach some rabbit brain processing power smartness(let say 1kg), it have to be 10'000'000 bigger. It's 10000 tonnes anthill, just in ants mass. (feeding and thinking capabilities proportional to small rabbit)
Funny thing, principles behind information processing, for most(if not all, I mean, I just doubt a little about bacteria's and viruses, but looks like the do) of creatures we observe on earth, are so much flexible, that it works with ant's even in case of such efficiency difference.
It works, but in same time it prevent's them to reach rabbit philosophic ability's(rabbits are hedonists - eat and propagate life, and they are good at that).
Mobility, another thing which makes brains and similar systems as useful thing. Constantly changing environment, changes are introduced at least trough moving in different parts of that environment, full of other moving things which wish to make energy from others moving things, and do not let to make energy from them self.
In this sense even mobile ants colonies, are not mobile. Rabbit is small and available information around him, is kinda small (field of vision(compact, eyes), sound which reaches place where he is(compact, ears), odors which reaches his place(compact, nose)) - so information is reduced, at point where it is collected in amounts it reach these points of information collection, and because of such reduction he have to think more to recreate important for him information which is behind these bit that reached him, but because of that fact it is reduced, he is capable to work with it, it fits in his small brains, would it be more, he would have difficult times just to filter that information, he could do that, but price will be loosing higher abstraction layers.
Ant-rabbit is way bigger then usual rabbit, and surrounded with tiny bits of information like white noise. Noise because it's not important, nor critical for him, and he is't capable to process it all, too much information. And because of size it easy may let him ignore it - eat what can be eaten, eat what moves and so on, on "reflex" level - no need to think.
Ant-rabbit as mobile and stationary system have major flaw in each state
As mobile system, because of it's size and weak structure (as whole structure strength, not as elements(ants) strengths, but binding energy between elements) can't be 3d structure, it have to be flat and this will reduce speed of his thinking drastically, also it will reduce theoretically possible complexity for it(it will be dumb as rock if short). Although it may exist in that form for migration as example and so on. It still can feed himself, react, protect - reflexes will still work, it do not need brains for that.
After he can dig-build-find anthill to reassembly 3d complex structure, and get his consciousness back. And there another problem how to feed himself in that situation. He is big. It needs lot of food, lot of water. Human need let's say 100kJ/kg per day (just for rough estimation, human needs 8-10MJ/day). Ants-rabbit may need something around 1/100 of it's mass food a day, that is 100 tonne per day. Grass grow 40 tonnes per $km^2$ per season, let say 100 days for simplicity, means $0.4 tonne/km^2$ so it means ants-rabbit needs $250km^2$ area just to feed himself daily, and 1.5-2-3-4 times more for purveyance for time when grass do not grow, in case to keep that place productive. Way to much for his elements, big for single organism (but not totally impossible Trembling Giant he is 43 hectares (106 acres))
Also Army ant, they do that not for fun. Also ants do not eat vegetables only, and there are lot of reasons, if they would then they easily would exterminate grass in their reach, but living pray is attracted but that grass (for different reasons) and that extends ants reach and energy collecting capabilities and prevents them from exterminating their food sources. Ant's surrounding is a big bait.
If ant-rabbit will keep photosynthesis plants to produce energy for him, still insane areas needed for such weak internal organisation.
Potencial barrier to get consciousness is so big, this may get it with higher probability and speed Slime mould solves maze puzzle
- Slime, may be used as media for ants to process information, and as symbiotic part of system may actually solve some problems in getting that consciousness-brains. Maybe even in natural way, after ant-rabbits will eat everything on planet except symbiotic parts. But will it satisfy you as ants consciousness.
As above there are some big problems to solve, and to get that possibility to solve them, ant-rabbit's have to eat and destroy everything except them selfs. Ant-rabbit have great advantages and great weaknesses, too polar in his strength-weakness and not capable to solve these tasks in it's initial form surrounded by other creatures.
On top of that, getting brains and consciousness is't easy task by itself, I mean to level, where you may solve problems which you do not have to solve, when it may solve problems which it created for itself. Any living creature, may and solve (as spices, and at some extend as individuals) problems which it have to solve. All these who did't, or failed at some point - are no longer living.
All that processes takes time, this is race, and cell have leg-up in that race. But even if we place on start line ants and cell - cell will win. Ants playing cell, have not chances to win race. And first ant-bear slime will rip Ants-rabbit apart, even for Ants-human, Ants-million-humans, Ants-1000millions-humans-pre-1963techlevel-science-approach it will be a problem.
Too fragile on each point in time, until it gets real smart and advanced tech.
Probably, ants-rabbit have to be created, to skip all that fragile development. Or environment have to be created to prevent breaking factors, but it will unpredictable limit abilities of such system to develop itself.
Being friable structure, it's very tricky to affect environment. I mean, elephant may move trees as whole, can ants do that, piece by piece yes, as whole very tricky, they may, but it will be already more complex task for them then for humans. (just imperfect example to show, idea)
Possible, but natural way highly unlikely.
Is there sense-point-value-meaning-significance-spirit in to creation of such systems? $Yes.$
We are pretty good at creating problems for us, and at solving them. But also we are limited in types, approaches, classes of such problems. And to be able breach boundaries of our collective consciousness, we have to solve new types of problems, and for that is't reasonable to create another creature with another types of problems, which will be not typical for us. It may create both types of problem have-to-solve and do-not-have-but-may to solve.
Besides, it's interesting task by itself.
If we survive, then in not so much distant future there(in this star system) will be no problems we can't solve (except our typical problems), and probably few more fundamental problems. And to get ability to solve more fundamental problems, and be better at solving our problems, it might be good time to get new toy, which will teach us by it's problems by it's existence, which will transform our ways of thinking farther. Next race we have to begin and to race.
Actually space may be good place to create such ants-brains. Less restriction from gravity, less energy to redistribute food-energy, more energy in general - all energy size restriction will be less strict.