I would move the whole thing up one size (not more) or separate the two plants, if it's not too hard. Not immediately, though. Wait a while and let them get adjusted to their current situation before you make any moves.
If and when you do unpot them, time it for when the soil is dry. Be extra gentle when handling the roots. Maybe shake off some of that soil if it comes off easily, because it's kind of rich. Use a mix with half rock (you can buy bagged cactus mix or add perlite/pumice to regular potting soil). Make sure the new pot has a hole at the bottom and ideally is wider than tall. And most importantly do not water immediately after potting up your aloes... wait a week or maybe more if you really injured them.
Your aloes will survive lots of sun but probably do much better for now given some protection. Especially overhead. If that balcony has a roof, great.