Making my rented apartment's Chamberlain smart?

Doug - You just need a way to open / close the door, and do not necessarily need to know whether the door is open or closed, correct?

The most renter friendly way is likely for you to buy a dummy door remote (look for “Security +”, not “Security + 2.0” remotes) for maybe $5 ~ $15, and program your remote to your opener, via the wall penal which you said you have easy access to. Lookup the manual of WD822KS and see how it is done - typically the procedure would be among the last few pages of the manual. No ladder needed.
(Or, you can ask your landlord for a spare remote.)

So, when you have 2 remotes in your hand. You can pop open one of those, and solder a dry contact relay to your remote bottom. Shell 1, 1 mini, uni, and a bunch of other ESP32 boards would all work - as long as it has a smart dry contact relay - and as long as either the smart relay has the feature to config, or you can automate, the press-and-then-release behavior to simulate human finger button press.

And then you will need to figure out a place to hide your contraption, and a way of powering (110VAC or 12VDC or 5VDC or whatever works on your remote) the whole thing.

For fun:

And I would not discount the idea of SwitchBot finger just yet. Yes, the finger bot units are bluetooth, but they also have a couple of different SwitchBot hubs - many if not all of them do wi-fi and can command multiple bots & sensors.