I have an old Sea Nymph riveted fishing boat that leaks a bit, after about 5 or 6 hours afloat it will get enough water in it to kick on an automatic bilge pump in the stern. I have used it like this since I got it without ever having any problems. I just put a new Seaflo automatic pump in last week, so I am all set for a few more years.
I had a Tracker pontoon that developed the exact leaks that Scott described, along the welds at the top of the logs where the M bracket attached. I drilled out 3/8 inch holes and drained it out. I then tapped the holes with a 1/8 npt tap and ran plastic pipe plugs back in. I ran the boat like that for most of a season, draining the tubes every other trip or so. This is a very thin wall and you don't get much of a thread in it, but it held for most of a year. I got the boat fixed the following year under warranty and the dealer put half couplings on the logs to put brass plugs in at that time.
I told both stories because I think the second method is a reasonable temporary fix for your problem. The permanent fix is a trip to the welder.
I also don't see any reason why you couldn't cut a small window high on the rear of a log and drop a wired auto bilge pump in to pump out the accumulated water. I won't pretend to be a pontoon builder but I saw the inside of a few that were opened up to shorten them and they have all been the same in a few ways.
They all had chambers that were walled off all the way from the top down to ALMOST the bottom, with a small gap to let the pressure equalize and let water in one section drain to the rear but to keep the chambers from all flooding if one suffered a major breach. The were also all vented to the outside in some way in at least one of those chambers. to allow for the changes in pressure from cool to hot as mentioned above. As they were constructed the only chamber that could flood completely would be the one with the breach and the one with the vent.
I don't see any reason why a bilge pump, dropped through a small window cut in the rear of a tube, and then pumping back through a thru hull fitting in the window cutout, wouldn't work just like the one in my Sea Nymph. You would have to pop rivet some pieces to the cutout section to allow you to put it back in but it should work. I would bet anything you want that water entering anywhere on the tube will find it's way to the lowest point in the tube eventually, just like my old Sea Nymph. That point is most likely the lower rear of the tube.
I thought of doing it with mine before I got it fixed but the drains were a much simpler fix. Just like someone with a leaky old boat who uses it all the time and pulls the plug out at the end of each trip.
I would never try to pump out a tube by putting air into it, I have heard too many stories of people blowing them open with low pressure, under 10 PSI. while trying to leak test them. I've seen pictures on other forums from manufacturers who showed the ripped open tube that resulted from someone mistakenly putting 15 PSI of air in a tube being tested, instead of 3 PSI
I would also make sure the tube was vented, high in a dry spot, if I put in a pump. but I don't think the little bilge pump would collapse the tube if it started to draw down a vacuum, it would more likely just lose it's prime and stop pumping.