Brandy,
Honeysuckle is a twining vine, so I am surprised that it will even grow on your vinyl lattice. I must have a smaller hole type than yours, because twining vines won't grow on mine.
Clematis on the other hand, loves lattice.
I put a pretty metal trellis in front of my lattice and planted Mandeville on it, (which is a twiner), and it loves it and is doing great!
On the vinyl lattice itself, I planted a clematis right next to it at the bottom, and it is climbing right up the lattice nicely.
Clematis is what I would go with if I were you.
It sounds as if you have Japanese Honeysuckle by the way, is it yellow and white? DOes it smell really good?
If so, get rid of it. It is too aggresive, unless you will not plant it near any other bushes at all, in that case, grow it on a piece of fence from a local hardware store, or buy a trellis for it. Japanese Honeysuckle is illegal to sell in 28 states, it kills native shrubs and trees.

Thank you, butterfly. I have had both of these honeysuckles for 5 or 6 yrs. Neither has proved invasive, though I am constantly pulling up native honeysuckle that creeps out from the woods around my yard. One of these does have the typical honeysuckle flowers, the other one has the tuberlar red blossoms. I have to thread the vines through the lattice, you're right they don't do that themselves. I think I will look for some of that green wire to put in place of the lattice and move it to the front deck where the clematis are. Brandy