I don't know what kind of honeysuckle you've got, but hey, it's a honeysuckle and can take a fair amount of abuse. I went at mine (a sempervirens, about 5 years in the ground here) from underneath - cut out the oldest branches, reined it in from my neighbor's side of the fence, and was pretty ruthless about the rest. I'd say that I reduced it, overall, by half. This was in May, and it's about as big as it was before, even with the heat and no rain.
You're in Zone 8, so you can probably do this anytime, as you won't have the winter downtime.
These are tough plants; don't worry so.

That's great! I'll be having at those vines then. I appreciate the advice!