Having just returned from the hospital, I haven't seen this thread before. The 375H&H is really a little too powerful for the typical 16 ga frame size many are made on. They pass proof testing, usually, so are considered safe. Continued use may bring up otherwise not noticed problems. Raimey's idea of disconnecting the safety makes sense, to see if recoil is causing the automatic safety to partially reset to "safe". Bushveld's discussion of chambering belted cartridges is great if you are building a new rifle, but the one under consideration is already chambered. For many years, I set the headspace in my loading dies by smoking the neck and shoulder of a case fired in the particular chamber and set the die to just "kiss" the shoulder of the fired case. This causes the handloads to headspace on the datum line rather than a belt or rim. On the other hand, I think this is a different problem having little to do with the heavy trigger pull. LeFusil's idea of recoil causing the sear to fall from full cock to the intercepting bent is worthy of consideration also. BTW, if setting the cartridge headspace with the sizing die, for a double rifle, the shortest chamber should be used to avoid having to load separate cartridges for each barrel.
Mike