I think you’re right that Respirocytes triggers with Levy AR Lab Access if it’s the first time you go down to 0 cards, just like Order of Sol during the resolution of Sure Gamble, since the wording is identical.
The wording on LARLA is:
Shuffle your grip and heap into your stack.
I don’t think you can break the shuffle down any further, and you would draw one card after the shuffle is complete. Then, draw 5 cards after for a grip of 6 cards at the end of the resolution. But, if LARLA is the last card in your grip, you would trigger to draw one card (if possible) then shuffle your whole grip and draw 5 and end up with 5 cards.