Thats why using voltage to check the battery
state of charge is a challenge. Voltage will drop when there is a load. Voltage will drop after the surface charge from charging dissipates (in a few hours). Nothing sounds unusual in your scenario. 12.5 is still about fully charged depending on when the reading is taken.
Instead of going by voltage alone, I keep track of current (used and replaced).
As for the pump, I am sure it can be done. Have not seen it myself. Check valves to protect the hand pump when the tank is pressurized should be used I think.
How many showers do you get from the onboard tank? Two maybe?