With my 2005 Diplomat and Michelin XZA2 tires, I run 95 PSI on the rear and 105 PSI on the front. After running for 30 minutes, my TPMS will usually show about 112 PSI rear and 122 PSI front. My Michelins are rated for 130 PSI cold.
You should gain about ten to twenty PSI when warm, depending upon ambient temperature. Your 105 and 109 PSI are slightly high, but should be fine for this coach. Your tires are rated for 120 PSI cold. I am slightly heavier on the rear axle than you are because we have the PAQ.
Yes, I agree that a four corner weight is the best solution.