Never one to pass up an interesting rabbit hole, I chased the "RECIRCULATING SHOWER" for about an hour.
My results are depicted below.
First my Observations
1. For now, Propane heating would be the easiest and most efficient setup. Here is an affordable Outdoor model ($200). It fires the propane off of 12v cigarete lighter, but other wired models are available. Relatively low power requirement.
https://m.media-amazon.com/images/I/51Yrxq+-jEL._AC_.jpg
2. Soap can be readily "scrubbed" and grey water systems are wide use for gardening. A DIY Filter can be made relatively easily:
DIY filter for grey water recycling:
If you'd rather buy one - even though looks simple they are asking a lot:
Compact Grey Water Filter:
Matala 10 Grey Water Filter - compact and efficient Pikkuvihreä but there are some good ideas there you could use/
3.UV purification is available in 12V but not cheap. NOTE they show a COLD WATER INPUT.
If that is a requirement - there is no recycing until the water cools. A means of storage and adequate supply for the shower on a "one-time" pass. A 5-6 gallon "Jerry Jug" might be sufficient.
4. Storage and assembly - my thoughts still lean towards a Bumper Storage Box, with all the gear inside. The OP mentioned
Infrequent or Occasional Use. The recycling remains an issue if the UV sanitizer needs cold water.
Here now in pictures:
I do have experience using a Propane fired "Instant Water Heater" similar the the unit shown. Was not used in a recirculating ststem, but the sketchs I've laid out show what might meet your desired objective.
Costs
12v UV Purifier $800 18Watts
Propane Heater $200 n/a
Dedicated Water Pump $50 36Watts (could elminate cost if bought with the heater.)
Bumper Storage $200
Approaching $1200 - Thats a lot of Baby Wipes. But, a novel idea.
Thanks for offering the challenge.
Cheers - Jim
BTW - Next we'd go for a Hydrogen Electrolyzer and elminate the propane!