Covering the drain should help, but won't stop the water from evaporating from the P-trap, as it is open on the other end to the waste water tank, which is vented. It will slow down the evaporation rate, but once it is evaporated, the odors will probably return.
if you want to try that, I suggest you use one of those large diameter flat drain covers/plugs - you know, the old "one size fits all drains" style. That way, you can easily remove it to add water to the trap should you notice odors again.
That said - I take showers in my TT anyway. I only have a 6-gallon heater, but the water is REALLY hot, so I don't need to use much to get a comfortably warm shower. Wish I had a tankless heater! But if you like to take looooong showers, then your waste water tank would be more of a constraint than HW capacity. I must say though, that's a really expensive laundry hamper!
2015 Jay Flight 32RLDS Elite
2012 Chevy 2500HD Crew Cab LTZ (6.6L Duramax/Allison)
Equalizer Hitch (1200# bars)