Pappy, obviously we live in way different areas so probably different plant life. I don't know if your weed is what we had last year. There was an odd plant that grew in the flower bed by the garage. At this time of year that side gets overgrown. Then we just clear it out in the Fall. But this plant grew so fast and climbed the garage almost overnight. It also wrapped around cable wires and everything in it's path. Like you say little sucker cups, it pulled paint off if you tried to rip it.
Finally a drastic measure...it had gone all ropey and thick, the higher it went the worse it was. This is not a method I recommend but it worked for us...
Use sharp clippers all around the connecting branches
A screwdriver or long sharp object to try and loosen roots
Take rope and perhaps bungee cord and tie all around the biggest weed parts
Connect securely to a sturdy vehicle
Then slooooowwwwleeee back out...the weed should come down in one huge piece.