This isn’t an issue with solo cups, literally hundreds of us use them without issue. Probably had more to do with using transparent cups in the beginning if anything, or maybe lack of oxygen to the root zone. Here is a link to some of the solo cup activities around here.
Only difference is that you’re new and most of the members from link aren’t. You just need more reading, learning, etc. We have all been there at one time or another.
Transplanting isn’t going to hurt you if done properly, so go ahead. But you’re going to want to get to the bottom of why your plants are so small at this age and correct whatever issues or you’ll run into same thing once in larger pot. I would start with environmentals, then look at plant care. What is good? Build on that and start fixing the things that aren’t. I would say simply from looks your light is probably too close or under powered.