> So I could 'legitimately' order char sui tsa sui?
Not really, for a couple of reasons.
The "r" in "char" is a rising tone, which is distinct and different from the "tsa" in "tsa sui" (Mandarin) or "tsap" in "tsap seui" (Cantonese).
> How am I pronouncing the 'tsa'?
The "tsa" and "tsap" are (and I can't confirm this 100% since I only speak rudimentary Mandarin) most likely sharp falling tones, like when you say "Shit!" If you say "cha" with that tone, odds are that it'll sound like "chop" with the p not being very strong. The closest approximation is probably saying something like "it's a" but without the "i" and in the tone/falling pitch of a curse, like "Fuck!"
They're very distinctly different in most Chinese dialects, and Cantonese and Mandarin are distinctly different dialects themselves.
The other reason you couldn't say "char sui tsa sui" is that "char sui" (correctly, "char siu") is from Cantonese, but "tsa sui" is from Mandarin, so you can't really use them together.
Another issue is that foods don't always work like that -- imagine trying to go to the pub and ask for "salmon and chips" someone unfamiliar with pub culture and English might imagine that it's just fish and chips with salmon being the type of fried fish specified (not knowing that you can't specify the fish), or someone else might expect grilled salmon with chips on the side, but odds are you might not get anything at all because it's not something anyone usually serves. :)