I don't know of many vines tropical or otherwise that are not aggressive. Tropical plants usually don't tolerate a hard freeze.
I am sorry to hear about your problem. I feel for you, I have California grass growing 15ft tall in my yard and I eventually had to deal with it with Round Up after trying and failing to try to keep hacking it down. It kept spreading and only with round up have I had any success. I am still working at it, but I am finally making headway. At least I am not stuck in the same corner trying to kill one clump while the rest of the weeds keep spreading all over the yard.
It looks like you will have to take a multi pronged approach. Use Round Up Ultra to kill patches and tilling, mowing to prevent the seeds from spreading. Your cold compost is one of the problems. You can't kill weed seeds that way. I would essentially get rid of the pile in its current location. Sift through it or dump it and start another pile somewhere else. It needs to be a hot pile to kill weeds so make the pile all in one day. Collect all your browns ahead of time. Make sure you have no seed heads in your greens. Build your pile 3x3x3 minimum layering greens and browns. To get it hot add urea or sulfate of amonia, and fresh manure in the center of the pile. Make sure each layer is watered to the consistency of a dried out sponge. Finish with a layer of browns and cover with a tarp to trap heat and in your case to keep weeds from sprouting on top. If the pile is made right and is tall enough, it should start cooking and steaming within 48 hours and can be turned after the 4th day. The tarp should be able to come off if the pile is heating correctly. The more turns, and with the pieces chopped or chipped fine, it is possible to get hot compost in a short time. It also helps if your day temperatures are ideal for growing as winter temperatures slows down the activity of the microbes.
My first attempts at composting was less than stellar. I had cold compost too because I had 90% greens and not enough browns. It started out tall enough, but with so much greens it became a stinking slimy mess very soon and the weeds just grew on it and through it. The weeds were the primary components in the pile anyway and these weeds can grow from a node. I finally went to bag composting instead. It was still slimy; stinky and quite anaerobic but it did kill the weeds and it did make decent compost. Since it was anaerobic anyway, I did not worry about turning. I just had to open the bags once they shrank and the weeds were dead so eventually they could dry out. After that I figured out since I had so many greens, that it was better for me to build the pile in a day. I could gather up all the browns all at one time to make the layers. It was the only time I really got a hot pile done right. I stopped putting perennial weeds in the pile and took those to the dump.
If you get really desperate, you can use a soil sterilent which will essentially prevent anything from growing for one year. At the end of the year, you will have to re amend and start over. That was the choice I had with nutsedge, but instead I decided to crowd it out instead. It is still a problem and became worse recently. I did not notice my sprinkler system had a problem until the grass started dying. I am still trying to get the grass reestablished and I am hand weeding and using sedge hammer, Round Up, and Image to try to stay ahead of the nut sedge.
Happy gardening in Hawaii. Gardens are where people grow.