This is going to take a little bit of creativity or custom work. That Robert axle is something I've never seen, but it looks hopeful.
I have two bike trailers. They're both made by Burley. One is from the 1990s, designed for carrying children, and similar to yours (except for the hitch). The other is of more recent manufacture and designed for cargo, with a hitch more similar to what you have. Neither of them would work well with every possible style of axle and brake, but there are always workarounds. I know that isn't really much help.
I'd say you could safely attach the trailer hitch to your chain stay, if you also fabricate an extension to the trailer's arm, to make clearance for the bike wheel. One thing to watch closely before attempting this solution is your heel clearance. Your chain stays are not very long, and anything you attach forward of the axle and brake could interfere with your pedaling.
You could also perhaps attach it to the seat stay, but will have to keep the attachment point as low as possible, so the trailer rides level.
The important thing in either case will be to protect the carbon fiber frame from damage due to chafing or localized pressure/stress. Even a thin structure like the seat stay can take quite a bit of force on it, but could be easily damaged by direct contact with metal parts or by rubbing/scraping. Again, that extended through-axle might be your best bet.
If I were choosing between those two bikes for towing a trailer, I'd probably go with the mountain bike. It likely has geometry that will work better for this purpose, and I suspect will be more comfortable for the type of ride you will do while towing your pup around. It's really up to you, though. Either one could feasibly go well, assuming the axle extension works.
If you're planning on riding steep hills, you should take the bike's gearing into consideration. Towing a ~70 lb. dog plus the weight of the trailer itself is not easy. I have hauled about that much weight behind a few different bikes. It is doable, but it adds a new dimension to the biking experience.
Last edited by Broctoon; 10-03-22 at 11:10 AM.