Add Left to Sub -3db
Add Right to Sub -3db (because apparently when you mix channels you need to take each down 3db)
This is probably part of your problem. A -3dB reduction is not enough to avoid clipping when summing channels in all (or most) cases. It's true that combining two perfectly uncorrelated signals sums +3dB; however, summing two perfectly correlated signals sums +6dB (which is why a linkwitz-riley crossover sums flat and a butterworth doesn't).
Most real world signals are somewhere between those two poles (not perfectly correlated, but not perfectly uncorrelated either). But you will always need a minimum of 3dB of attenuation to avoid clipping when summing channels, and except when summing uncorrelated noise you will need more. 4.5 dB is a compromise position that won't guarantee no clipping, but will do a much better job than 3dB. If you want to be perfectly safe, 6dB is the answer.
Additionally, I think the "add" reduction may only reduce the volume of the channel being added, not the destination channel, but I'd need to check on that to be sure (I'm not in front of JRiver right now). It's worth investigating to make sure though, because if that were so you'd have lots of clipping even with uncorrelated noise.
You mentioned room correction above, do you have any settings in that DSP module?
All that said, here's the good news: none of that will matter if you use internal volume unless you have the volume maxed out. JRiver's DSP stack uses any headroom created by the internal volume setting, which allows signals that would clip at full volume to be represented without clipping or additional attenuation. So if you have internal volume at 100%, and clip protection engages and cuts the volume by 3dB, if you have the internal volume set to -3dB or below, clip protection will never engage because the necessary 3dB of headroom is already there. Also if you need to add boost to the chain for some reason, you can just make sure to set your maximum internal volume so you don't compromise your headroom "envelope."
If you use other volume control methods, clip protection will engage and turn down the volume anytime a clip
could happen. That's probably what's currently happening in your setup when you see clipping, and why it doesn't sound like anything. Watch the "audio path" while it's happening and see if clip protection shows up.