I would probably leave them for several years (4-5 minimum) before making that decision. Clematis only start looking good in 2 or 3 years IME and then get fuller and more floriferous after that. If they are happy, they will fill in well, and IME the honeysuckle (I'm guessing you have a cultivar of Lonicera sempervirens, a native) will expand quite a bit and may crawl over to the top of the clematis trellises. If you add some perennials in front of the clematis that will fill in for the lower leaves that sometimes drop off type 2 clematis such as Ramona, I think you will be pleased with the overall result. If you haven't ever hard pruned Ramona, you can do that this winter. You will delay the flowering, but it should fill in more due to the pruning encouraging additional branching. You could instead get one of the type 3 clematis since they often hold on to leaves better IME.

I honestly can't think of another vine that will do what you want here without being a pain to control. You want to avoid things like trumpet vine and wisteria which are both too large and tend to send up suckers and/or seed around vigorously. Both climbing hydrangea and dutchman's pipe vine are also much too large (20-40') for your garden. I don't think you have the space for grapes or hardy kiwis either though you could consider training a kiwi along the chain link behind the trellises if it's yours or you can get permission from your neighbor. Annual vines will take a while to fill in each year, longer than the clematis will.

I had another thought of something you could add to the trellis without removing the clematis. Look at the climbing rose Zephirine Drouhin which in my garden is virtually thornless, blooms well, resprouts from roots quickly (since it dies back to the snow line here), and blooms spring and fall, just not in the hottest part of the summer. Nice scent, too. It won't have really dense leaves, but the combination of the clematis and the rose may have the density you want. You will need to fasten it to the trellis with clips or ties since it doesn't twine like the clematis and be sure to get one on its own roots so if it does die back it resprouts true to type. I got mine at Brushwood. They say zone 5, and mine has survived -23 without problems.

I really love that idea! The color combo would be beautiful, and a rose without thorns- yay!

They may bloom together or may not. I planted my ZD with the early summer blooming Prince Charles, and ZD often isn't blooming much when PC is. I still really enjoy both of them.