Can I put a breathable peel and stick weather (air) barrier on the inside walls and ceiling?

There would be no problem adding a peel-and-stick membrane that covers interior spray foam, and it would definitely give you an excellent air seal.

But a 'double vapor barrier' isn't really something you need to concern yourself with, so don't go out of your way to find a breathable membrane. Even if the spray foam itself is not providing a proper air seal, it is most definitely acting as a vapor barrier, so putting another vapor impermeable membrane (like 6 mil poly) right against it really wouldn't make a difference in terms of vapor migration through the wall and roof assembly.

And while that solution would work fine, it sounds like an expensive one. You could save quite a bit of money if you were to just use standard 6 mil polyethene. If you pre-seal the joints in the poly with acoustic seal and even cover the staple holes, I think you could achieve a comparable air barrier but at a fraction of the price.

My only concern in what you write is not knowing exactly what you mean when you say the seams are 'thin'. If you mean there is an insufficient amount of insulation that is causing the problem, then you will be experiencing a lot of heat loss, if that's the case you may wish the thicken the insulation for energy efficiency purposes.