I also bill pay with Chase and they debit my account as soon as the check is issued. I've not run into your problem (that I know of) but the UCC says
A bank is under no obligation to a customer having a checking account to pay a check, other than a certified check, which is presented more than six months after its date, but it may charge its customer’s account for a payment made thereafter in good faith.
So a bank may honor checks after 6 months if it wants to but that's the point it becomes stale. If Chase is crediting your account after 3 months, it doesn't mean your check is stale because it isn't. They're just making up for their at the time of check issue debit, which is actually premature since its not cashed yet.
When your LL cashes your check, Chase will debit you just as they would for a normal check. If 6 months go by and it becomes stale and Chase doesn't honor it, I would think you're ok since you paid your rent by check in good faith. There's no ill intent. You'd just reissue another check. (I say "I would think" bc it's been since Reagan that I took UCC but good faith + intent + his fault for not cashing before staleness = you're probably good)