I`ll try to help out here. Firstly, congrats your roots are growing quickly! As for the size of the pot you should be upgrading to all depends on how long you want to veg your plants. A good rule of thumb is to have 1 Gallon per vertical foot of growth. Given the amount of space you`re limited to, I`d say 1 gallon pot should be fine, or 1.5 gal to be safe. You can transfer the peat pots to the new larger pot full of your New potting mix (ie: potting soil, peat, perlite, etc...) Just be gentle taking the peat pots out, or better yet scoop around them as to not interfere with the roots around the pots.
PS: Square pots are the most efficient at utilizing space