They won't do on a trellis, since a trellis and vines will not support the weight of the melons. You can prune a couple of vines out at the base and leave them to die back instead of trying to pull them out of the way. The vines are already so intertwined that you will damage the vines that you want to keep. If you planted them four to six feet apart, they probably have enough room, and you can prune longer vines to a manageable length. You can also prune off some of the smaller melons so that the plants will spend their energy on the fruit they are already producing.

I am having the same problem with ornamental squash vines overtaking my cucumbers. I suggested to DW that I prune the squash, and she said no, she wanted the gourds for her classroom and would do without the cukes.

Best wishes

if you have a STRONG trellis, you can grow honeydews and the small 'sugar baby' melons that way- but it involves making slings for the fruit, which is fun, but time consuming (it is fun though- and one of those plastic mesh scrubby bath things will yeild enough mesh to support half a dozen melons...the braided rope back scrubber kinds are even better deals)

i thin or prune all my vine crops. tomatoes, cucks, melons and squash, at least the ones i trellis. i trellis any melon that is the size of a sugar baby. i trellis my indeterminate tomaotes, cucks, small vine type squash up to butternut size and melons up to sugar baby size. pruning allows the plant to put it's energy into the fruit instead of the vine. i have a pic of a 15 pound butternut squash 4' foot above the ground. trellising allows me to put more plants in the space i have. this year i have about 6000 sq feet of garden pace. there are 54 rows at 40 feet long. an example of my intensive gardening i have 9 rows of tomatoes and i have over 500 tomato plants. the trick is to feed the plants and pour the water to them.
good luck with you melons.
WiringMan