Here in the south, Japanese honeysuckle is a huge problem (we don't have many of the shrub honeysuckles). I say forget the deer and take care of it yourself. It's not just that it will strangle the trees, but it will create more seeds and spread further.

I find it easy to pull them out after a rain (but problem is that I have SO much it is hard to make progress). You could try pulling as much as you can and then spraying any remainder that sprouts next year with brush killer.

In the south it is evergreen, so it is one of my "favorite" winter activities to pull it out when I have very little other garden work.

Japanese Honeysuckle is virtually evergreen here in NY also. Cut it, poison it, pull it, get rid of it. Don't depend on the deer. For whatever reasons, the deer don't seem to be denting it much (they seem to prefer whatever we have actually spent money on). Getting the whole root is tricky in that the vine spreads along the ground and can re-root in several locations along the way. It is easier after a rain. I would cut it out of the trees, pull up what you can and spray the rest.