I wouldn't, I don't, and don't wish problems for other folks. I've wept, rolled eyes, tried to help with folks who've gone 'outside' the main 'network' room with Blitz H preamps and controllers... Keep it as simple as possible. Keep the heart of your system near at hand! That forces virtually everything else to also be extremely weather proofed, etc. Plus you no longer have immediate access to the controller. You may wish someday it was next to your 'computer' setup... you don't have to fool with things like hooking it to WiFi, other troublesome 'expansions'...
Having said that, I know several operators have had good luck remotely. But I know MORE who've wrestled with various issues, such as encapsulation, water damage, communication, etc... Your H field antennas don't need to be outside, just away from noise sources.
E probe DOES need to be outside, away from noise and objects etc. That leaves you with 1 Coax to E probe to run outside. You can also, since you're renting and possibly may move again, view E probe as an option, disable it, and forget the coax, operating H channels only.
I sneaked all my outside cables (Cams, Blitz Es, etc) through an HVAC floor vent hole, and out through the crawl space. No Problem.
At worst, you only need the E probe coax to run outside!
Have you discussed your issue with the landlord? Mike