Hi there! The good news is that I'm pretty sure there are seats that will fit there. The bad news is that they're a little on the pricier side, mostly, and I wouldn't recommend a convertible from birth.
I would start by trying a Cybex Aton 2 or a Clek Lling there, as they are usually quite compact. Buy from somewhere with a good return policy like Amazon. If necessary, you can use a rear-facing only seat baseless to save room (the Clek Llingo might be a good choice if that ends up being the option that works- it's a seat designed without a base and it is made to be easy to install baseless. Also a bit cheaper than the Lling.)
Once baby has outgrown that and can be at a more upright angle, there are several convertibles that may well work.