You're right that it is a Honeysuckle. It will almost certainly survive being cut right back to a stump so you can repair the fence.

What color are the blossoms? If it isn't Japanese Honeysuckle do you ever do plant trades?

Certainly looks like Japanese Honeysuckle, and it grows so fast that I agree with floral.
Cut it to the stump They grow quickly, and may bloom a year from now, if not by Fall.
If it is another sort, I would be surprised, but then you might have a gem.
The Japanese sort is a terrible pest, here, and always apparent. I cannot, really, keep ahead of it, while I do, consistently, pull or cut it. Among too many valuable plants for a killer spray.

Just another word, if you choose to get rid of it. I had a magnificent Trumpet Vine. A perfect beauty, especially since it was on the pool deck fence. And low enough that I could take away any pods that might seed about. But it started to sucker, and, occasionally, after 3 years or it being killed, I may still see some. Every bit the monster I was warned about, but I resisted. I cut it, drilled holes in the stump, poured in serious plant killer spray and let the liquid soak in. Now replaced with climbing Roses, that have blooms, Spring to frost.

The Trumpet Vine 'New Jersey Yellow' at a time when I had just put in a new pool liner and was refilling the pool. The Rose 'Winner's Circle' is as disease and aphid resistant as 'Knock Out Roses' but is a climber.

With full Sun one sort of Honeysuckle, L. sempervirens 'Magnifica' blooms around July 4, scent-lessly and is a mess of brown twigs all Winter, but any of that species beats our Japanese Honeysuckle as a garden addition. Recurrent slight bloom, until frost.

Thank you for all of the comments. The snow and ice has kept me from it but I think I'm going to make a play for cutting it back in a week or so when the area dries and I'm able to get back there. I believe the blossoms are yellow.