TPMS - Newbie worried about what it shows.... is it OK?

All normal. Kudos for inflating to PSI based on the weight of your rig. Expect 10-15# increases as you drive. Ambient temps affect your tires same as it does you. Outside tires usually a bit warmer than inside. My fronts, 315/22.5 increase from 110 to 125 generally, esp on hot days. Remember those tires are rotating on a platform that runs about 160 degrees on a hot day. IMO, your readings are correct and within tolerances. I've run TMPS for over 8 years, both on the coach and the toad....Dennis