That staccato at about 2:00 actually sounds intentional; you really did a good job of covering all the little hiccups and errors, because it sounds perfect to me, even as someone who's played piano for about half his life. I think you should stick with this (and stick with the staccato at 2:00, because it actually sounds really good like that) as your final product.
Get that sustain pedal fixed anyway, though (it should be easy enough to fix on your own, though - something might be loose; I get the same issue with mine, and all I need to do to fix it is to just tighten a single nut).