I hate to say it but you might never rid your yard of this vine. I'd been trying to kill them for about 8 years and finally gave up. Now I just try to keep it confined to the far corner. It really is beautiful when in bloom. I've tried pulling it up, herbicides, including Vine-X, with no luck. I've even injected the herbicide directly into the stump. Maybe a few gallons of gasoline and a match? The trumpet vines and cockroaches will be all that's left after a nuclear blast.lol
Karyn

I don't know if this will help you with your TC, but maybe you could try this or a similar idea, and secure it somehow from the dog.
I had a Virginia Creeper, which I love in the right environment, but it was about to overtake my holly trees.
First I cut down all the vines I could and removed whatever roots I could. As soon as new leaves sprouted from near the ground, I used a paintbrush (NOT spray!) to apply the liquid Roundup directly onto the VC's leaves and stems, then covered it with a flowerpot held down with a brick, so that it could not touch anything else.
Later that summer, a few more volunteers came up from nearby roots, and with the same treatment, the VC eventually surrendered.

Bluebars, was your trumpet vine for sure dead after all of that. My husband and I have one we are about to tackle this summer. I am sad that I have to kill it because it is beautiful but it is getting worse and worse. We have much too small of a yard to have such a beast, let alone three of them. Your idea is exactly what we felt we were going to try but I would love to hear that you won. What a depressing thing, Karyn, to try and try and never win. Why is that? My biggest worry is that we are going to take them off the fence, dig out the roots of the three main parents, paint the other sprouts and then nothing will stop it and we have lost what was good about them...the flowers. What do you guys think?

It takes more than one applicaiton but persistence will do it. If you keep cutting it back and applying roundup, it will eventually feel unwanted.

I've never had any luck with Round-Up on Trumpet vines or brambles, even with repeated applications and injecting directly into cut stems. I've heard very good things about a herbicide called Vine-X but haven't tried it yet. I've finally found a way to coexist with the Trumpet vine but not the brambles! I have to get some and see if it's as effective as I've heard.
Karyn