All you had to say was "chicken coop door".
Anywy, these
https://www.texasindustrialelectric.com/relays_0332209150.asp or equlivelent relays are available cheaply. They need about 200 mA for the coil and that's all you need for a limit switch (>~200 mA and 12 V DC).
By cheap:
**broken link removed**
You can also get a dual socket for them with wires attached.
On the above page there's a link for wiring diagrams and common uses. What's at the bottom of page PDF/Real page 15 and top of 16 is the way (pick one) you need to wire them.
OK!
You need some sort of contact that is OPEN at each limit. Environmentally, your right it may cost you, but the contacts won;t see the full motor current just about 160 mA at 12 V.
Using Amazon as my unfavorite etailer, I find something like this:
https://www.amazon.com/Apex-Curtain..._SR160,160_&psc=1&refRID=SYK43727961WG8YN8CT7
Costly - Yes. With some flex conduit it's water and animal proof. There has to be other options. We have no idea what you have to work with.
You might be able to put a pipe on the door and enclose a non environmentally rated switch behind a plexiglass panel covering the studs. Fix the switches and use step collars for adjustments. It's just an idea.
The limit relays will not draw power at the limits.
One's elegant. One's quick and dirty.
You then need a relay that is connected to your timer. The timer is not fully explained.If your timer has SPDT contacts your totally set. Again, they will only have to handle about 200 mA. You get to define what on does (Open or close).
If we have to cut the power consumption, we can do that. Need more timer info.
So, maybe little more info about the timer and picture of the door so we have more to work with.
Aside:
There is a missing piece of the puzzle. Diodes should go reverse biased across the relay coils. These would protect your 12 V timer and there's a preferred way (circuit) to where you wire 12 V. Supposedly, the relays can be bought with internal diodes, The easiest, would probably be soldering them direct to the relay pins.