I don't have any experience with a real LA2A, so I can't really answer your question about comparing the sound of various types of caps. Just looking at the circuit, I think you could use just about anything you wanted for C13, C12, and the neon bypass cap.
I think if you're going to hear anything from caps it will be C1, C2, C3, and C5. In the original and reissues C5 is an electrolytic. I used a film cap that I got surplus. I believe it is metallized polypropylene. I didn't compare it to an electrolytic. To my ears, my Bloo kit sounds very nice.
Some of the other guys here have far more experience with LA2As than I do. Hopefully some of them will chime in. I'm curious about some of the same questions you are asking.
Analog Packrat