Nice question, I like the scale, but I would say it requires some better technologies for making K2, which with generative design systems and resources (computing power) when done on K1 scale, should not be a problem. In that sense this over bloated answer may give some ideas as well How can I move a planet? - it quite relevant and it uses technology one needs for the case, however technology itself may be not that well explained.
Things to note, if you cap material transfer to 10 billion tons it increases construction time much more, with the numbers in q - it is about each last 2h of work of 28trillion ton setup adds a year, so last day alone adds 12 years, or doubles the time. But with the cap and mass given the time can be calculated directly, leave it to you.
10 billion ton per second
Mars equator is about 20000km, and 10 billion ton is a ring on this equator 150m wide, 1m thick, and 20 million meter long (500t per meter of length, 3 t/m3) - each that ring should separate each second. It plenty, a lot, but it still in the realm of mass drivers.
Plasmafication is possible, but one of the biggest problem here is the loss of energy and a low efficiency of overall process. Making a planet go, even for a full K2 is a load, do not recall number for mars but venus earth it 7 days for 100% efficient system. Plasma will cool(emit waste energy) down fast because of ~T^4 and you can not heat the thing directly more than 6000K so there will be a conversion toll as well, and you will dream about even 1% efficiency, and more so with your partial K1 installation. So you have to strive for certain degree of efficiency, where 1% is your bottom line, 10% desirable, anything better is great.
You can't think you can throw energy just because you have plenty of it and that plenty may overcome all the problems. First of all until you have K2, you do not have that plenty of energy. Second of all, when you deal with planet sizes stuff, even smaller ones, it turns out K2 isn't that much plenty, I mean it sure a good deal of energy, but it not enough for everything.
So plasma thing will emit ligth(heat, waste the enrgy) if you heat it to 6000K(which may be not hot enough even for mars), at the same intensity the sun does on its "surface", and all that is a loss of energy in a plasma case.
So making K2 do requires a better technology(or longer time frame), but also true it still does not require magic technology or any clarck technology. We may undertand what we may need for making K2, which key technology we may need, it just matter of crunching numbers and thinking how to manufacture/start it, make it work and improve. That 2d nanomaterial - smart matter. It does not have to be that such nano, and more a micro as if name is a concern. And maybe not the only way, but one of low hanging fruits, especially for K1 starting point.
Without smart matter, 10billion is about the limit where conventional mass drivers can get you, and some equaterial strip of those, idk 100-1000km wide may deliver that level of material supply. (It needs to check if heating cooling may become a problem, this heating cooling will be the main limiting factor, and energy supply - whatever hits the bottom first)
But for armrace supply bottlenecks won't be a problem, but who controls the supply line will be what it is all about(we do not have all that many planets to pick from, they amy as well be in a similar situation) and reaching new level of technologies is another direction - so power competition on one hand(who controls the sources and installations), and brain competition as a second direction supported by K1 computing. With K1 as the starting point I would bet all my money on the smart ones, they will win the race.
So I would expect that each big power may have its own K1 setup(one of the possibilities), as it better if each has it than nobody has it. As hundred nukes may render mars ring system useless, break down, meaning if there is no disparity in a space capacity it relatively easy to disrupt efforts of others. Sure not everyone has it, it like great powers, after there is some numbers they may supress newcomers, and it will be hard to get in without a submission, concession, aliance. Laying it loosely on our current political reality it about 2-3 such installatins if they come to it almost simultaniously. Or one if someone actively jumps out and forcibly supresses the rest. And in those cases all the actions are not around installations but on the home planet, until space habitats grow as power and then they can supress homeworld. So there is plenty of space for actions, all kinds of development possible, but it not necessarly around the setup itself, which delivers materials, but more in virtual space of developing technologies, politics, science, alliances - all the places where people are.
additional note, deployment
Two things here - moving a planet masses around is not such a small deal, and reactive propulsion won't be a great choice, not impossible but quite a waste of energy or mass or both.
More importantly, you do get higher efficiency further away is the orbit. So that moving closer to sun do waste your energy and resources and brings you worse efficiency.
Making the thing at mars orbit distnace is acceptable. There are 3 components in the system - a reflector which concentrates scattered energy, actual things which do conversion of that energy if you need it, and radiators, your cold end of heat machine.
Reflector is thin, so it does not require that much material, cold end also can be some sort of foil, maybe the same reflector material - so being further does not increase that much the mass of installation, but gives yoh a colder end. Not necessarly a huge difference between mars earth venus, but probably noticeable one if you place the stuff at mercury orbit, do not think it worth it this way to make that long transfer in this case.
Another thing is that transfering energy at such scales and places is a important aspect, and kinetic way to do so seems a reasonable answer, if technology is up to that.
All that does require considering momentum and impulse conservation, on those scales it not a small deal.
- Can'tanswer in comment section, as my browser seems too old for modern day SE java script, or whatever, so do it this way.
If each of them is in their home system, then with basic preparations they are quite set to repel K2 backed invasion even with 1% of K2, if their tech levels are more or less comparable otherwise. In case if a smart matter is not possible, as an example and they are stuck to previous gen technologies stack. As transfering sufficiently big invasion forces to supress other system, and arrive in some reasonable time (fraction of ligth speeds) with no invisibility in space and interstellar medium(hydrogen and dust stuff) is one of the things K2 power alone may be not so sufficient.
So it then even more so a case of cold war and who gets to next level tech first, who is the smart, who is the good boy. And then even if one is K2 and another are K1 but with smart matter - K1 has good chances, not even talking they have then the means to become K2 soon (but they may prefer to keep it secret, so, they may be reason for them not to show atypical growth)
But okay it a different tricky topic of interstellar warfare, and space warfare which is a deep rabbit hole, and quite pointless how it depicted in movies, or most of scify stories, and it requires, just demands, a certain tech levels.
Which K1 can easily achive, in my opinion, but it hard to create an advantage in another star system even with K2 - just for reason it requires a lot of energy to just deploy/deliver means of attack, and for them it is cheap to set means of defence. 0.1c attack speed vs 30km/s deplyment of defence means is a difference of 10^6 times of energy spent. And attacking side has to overcome the difference. Hacking and alike, sneak invasions, inside job etc may be a better way to invade.
Dyson beams, eh, so so - never bothered to consider their potencial efficiency at such distances (ly's) but one gets 10% of K2 at best with lasers and they have physical limitations how good a beam you may have, meaning density of spot at arrivial. And as an example a 2 ligth second spot is not sufficiently strong to kill random targest blindly all over the system and that is K2 energy.
Relativistic kinetic stuf is better, but it easier to detect them from the distance, and more importantly deploy means of their destruction (some dust clouds, foil sheet, etc, which works for a beam as well) so it also not a wunderwaffe. But okay it offtopic for this q. You may look at some of the other answers of mine https://worldbuilding.stackexchange.com/search?q=user%3A20315+realistic there are some
Specifically this one: How does a spacecraft attempt an intercept course with a hostile one realistically (Part II)? read the premise of the guy, in that one I tried to look at how a more or less a proper star system defence system should look like, and it decent as for internal law enforcement so as defence against incoming forces - fusion tech, nothing fancy like a smart matter. This one as well similar topic Fleet tactics & strategies in a hard sci-fi setting with near light speed weapons, ultra hypervelocity kinetics, realistic sensors, and torch ships and there can be more of those mines and other ppl.