David, your theory sounds plausible to me.

I doubt that the increased intestinal permeability associated with SIBO would be large enough to allow whole bacteria to pass from the gut into the blood but it could well allow whole proteins from gut bacteria to do that. The immune response arising from that could lead to adverse side-effects, including inflammatory responses.

Has your SIBO been investigated? My understanding is that treatment with probiotic bacteria (often various kinds of Lactobacillus) is the first thing to try. If that does not work then antibiotic treatment can be considered.