Your idea is not the first and will not be the last either.
You could consider using heating with wood instead of electric. No need for heavy battery pack. Free hot water if you attach a water tank into the chimney 
Works offgrid everytime you want. Even condensation won’t be much of an issue if you build good enough ventilation and use insulation that doesn’t collect moisture… what ever that is in english 
Tent saunas have fireplaces about the size you would need:
https://www.youtube.com/watch?v=n11dYnJqUqY
Here’s one build log from travel trailer to moving sauna:
I’m pretty sure you’ll need to apply some google translate on it though.
Just replace dedicated dressing space with bed and kitchenette.
Here’s ad for a factory built one
https://www.tori.fi/paijat-hame/Saunavaunu_tupakeittiolla_62515786.htm?ca=18&w=3
Actually there is some search words at the end of that ad:
Kärrysauna
Mobiilisauna
Saunavaunu
Saunakärry
basically they are all just different ways to say sauna on a trailer… Look them up in youtube and your favorite search engine.
Telttasauna (tent sauna) - that’s what I used in the example above. Fireplaces meant for tents and boats are small and suite quite well for trailer built too.
If you wish to go electric anyways, well… build ideas still work.