It sounds like the OP bought three unrelated tickets. Skipping the leg to LAX wouldn't have any impact on the unrelated LA to Australia ticket.

On a stopover (>24 hours), bags are always available to the passenger at the intermediate point.

On a connection(<24 hours), you're asking for a "short check", but most airlines are cool about it on an overnight connection. Short-checking has become more of a topic in recent years because people ask for it in relation to a hidden-city ticket. That's where they book an A-B-C itin because it is substantially cheaper than an A-B itin. Airlines don't like this and they know the airports most associated with the practice. But your route clearly isn't one of these, so your request for your bag in Cancun would probably be honored.

I've never been denied a short-check request on an overnight connection. But I've never flown Interjet so you might want to read their specific procedures (if you can find them) or see if you can find experiences from people who have flown them.