Here is my shower set up! It works great!
4” sewer pipe and fittings
Bicycle stem
20psi pressure control on cap
Small tire inflator
Midwest Control CPR-25 Pressure… https://www.amazon.com/dp/B01KO7NVYK?ref=ppx_pop_mob_ap_share
AirXwills Electric Bike Pump -… https://www.amazon.com/dp/B08CXVBKTZ?ref=ppx_pop_mob_ap_share
*this pump appears to be better than the most. I did purchase another pump, but failed after a few trial. This pump is made of aluminum and strong.
Doing something similar but with 6” PVC inside the camper.
Do you plan on heating the water? I have something similar to @Taka and it heats up to ~75F in the summer sun.
@SLCRex, I am thinking of adding something like this…
DERNORD 12V 600W Immersion Heater Submersible Water Heater Element Stainless Steel Heating Element with 1 Inch NPSM Fitting Double U Type https://a.co/d/3mq2cSM
Heat, pressure and PVC don’t go well together
@Sunsetsearider, I have a pressure control with 20psi, PVC state that it can hold as low as 73psi, so I assume what I am using for is none issue, then on temp, I will not be doing anything close to be boiling, so I am pretty sure my usage will be considered as “light use”.
Agreed. 15-20 psi is all that is needed. That and a little help from gravity. 
@cvollers, yep…I don’t think people understands the fact of the same PVC pipes are used in residential buildings… If my simple usage can hurt these things, then all kitchen in residential homes needs a warning label stating “don’t dump boiling hot water in a drain” or “maximum drain usage allowed is 2 including toilet”…. 
PVC is tough stuff…made tougher in tubular format.
Your home isn’t pressurized or holding hot water. Someone has written it up with the specs on both, shown its beyond the claimed abilities and could possibly fail. Will it, probably/maybe not. Are there specs that say it can? Most definitely.
I used ABS for my 4.5" x 6’ long shower tube. Mine is heated by the sun, usually gets up to mid 70F, which is enough for us. I chose ABS because it is more shatter resistant than PVC. The amount of heat that PVC and ABS can handle under pressure is the same at 100F (38C). Without pressure, the ABS can handle 180F vs 104F for PVC.
https://www.engineeringtoolbox.com/plastic-pipes-operating-pressure-d_1621.html
I would exercise caution with using a heating element, especially with pressure.
@SLCRex, thanks! I am sure with pressure meaning minimum tolerance, so I am guessing here, but <20psi will be none issue with 100F ish temp. Also, it is very difficult for sun to heat the water in PVC, so I am pretty sure with pressure control, I’ll have no issue.
Apologies to all, I assumed PVC, but Home Depot sewer pipe & fittings are ABS as shown below.
https://www.homedepot.com/p/VPC-4-in-x-24-in-Plastic-ABS-Pipe-1204/202300521
great design man!!! I like the idea of using the bike pump for pressure. I have been tossing a few designs around for a portable shower setup but I kind of like this one.
Here’s a YT vid that shows how to make one:
