Right -- your setup uses so much source strength that any background should be real minimal. You can do this because an ion chamber doesn't have things like dead time (that I know of).
Wouldn't hurt to check one without a source, though. A cosmic can really dump some energy into things, particularly if a shower from the primary hitting an air atom, or something in the house roof. I can see that here with scint counters that can time-resolve the individual parts of a shower if it's up in the air far enough, as the products take different lengths of transit time to my devices. When I see coincidence from two that are across the room from each other, that's pretty much got to be what is happening. Now, this is with very fast phototubes and a fast scintillator (plastic) that together will resolve around 5 ns. A geiger tube will just count, or not count, once, on things like that unless the shower was produced really far away (out in space or something) as the time-spread of most showers is well inside the response time of a tube. The ion chamber, by it's nature, should have those issues at all at any reasonable rate -- in a real extreme, I guess you can't ionize the same gas twice, but that's not real likely!
Unless -- and this may not pass some giggle test -- the energy dumped into a source atom(s) is enough to make it decay. For example, it's known you can fission uranium with hot enough gammas (or anything else hot enough). Since there would be energy gain involved here, that might be enough to cause what you see? This is pretty far out on the skinny part of the tree limb...but who knows?
Cosmics have plenty of energy to create multiple particles in a shower with enough apiece to do things like that - a lot of early high energy research, including discovering the positron and some muons and so one was done with them, since there were no accelerators that good at the time.
Another example of this is
Carlos Rubbia's "energy amplifier" where he proposes using an accelerator to burn up nuclear waste. Even with the inefficiency of a big particle accelerator, he predicts net energy gain. Could something like that be happening here? Not a clue, but it should be possible to find out.
A little digging might turn up some type of isotope that doesn't self-decay, but will fairly easily when hit hard enough. Bismuth might do it for example (I'll have to get out the books). I know that if I put a couple inches of lead over a counter, I get *more* counts when primary cosmics hit it from showers produced in the lead --but fewer from air-showers (because they have low enough energy for the lead to stop them), so that effect is kinda hard to tease out. Maybe I need to put a scint detector in my "cave" in the back yard and look at the energies of what it sees in there (about 20 feet of rock roof). In this case, the rock itself is a little radioactive -- you can see it on a survey meter, but I'd bet it really cuts down the super high energy cosmics and just shows a U + daughters spectrum -- but a test beats a guess anytime.
Posting as just me, not as the forum owner. Everything I say is "in my opinion" and YMMV -- which should go for everyone without saying.