Where my trumpet vines were coming up was mixed in between peonies and daylilies on both sides of the fence, and a row of hostas, bugleweed and some phlox planted on my side. Using the targeted application of glyphosate on just the cut stems hasn't damaged anything growing with the vines, and it's really mixed in everywhere. You do have to be very careful to not spill any, apply only a dab to the open cut, I don't even let a drop fall onto the soil or any other leaves, and this year I'm finally seeing a lot of progress in eliminating it. I think I only applied it to a few plants early summer and haven't seen anything since this year.
It can be done, but it's really tedious and you have to commit to keep after it. There were many times I really wanted to just give up in the summer heat trying to find them at the soil level and painting the poison on the cut stems. When they're growing in between other plants, once you cut it, it's hard to find again to apply the poison. I started putting my pruners with the tips pointing directly to the stems to help me find them again.
But if you have the space for trumpet vines, not on a fence or your house (it'll eventually bring down any fence or trellis), I'd say go for it. Be warned they will be sending up suckers all over the areas surrounding the vines including in your grass. About 1/2 mile from me a house has 5 of them shaped into trees. If they all mow at the same time, you see hundreds and hundreds of suckers coming up in the grass not only on their lot, but their neighbor's lawn on both sides as well.
