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.