Hi
Depending on what time of day / night you're looking to do this - some ideas might be:
- leisure centres (changing facilities usually have shower facilities),
sports clubs (if you tell them when you'll be passing beforehand so they know about it),
some youth hostels might oblige - again you'd probably need to make contact before hand.
Garden Centres often have separate toilets for people with disabilities which have privacy and wash facilities - as long as you're not depriving the people who need them, from using them for an unreasonable period of time. You need to ask about in the communities you're planning to pass through... ideally before you set out.
if you're riding in summer you may find campsites will allow you to use their facilities too.
Bearing in mind there's a cost to providing and maintaining these facilities it is, I think, good manners to make a purchase in café / put money in donation box / offer to pay
Microfibre towels are quite light weight...you don't need to take a
whole bottle of shampoo with you...and several riders of my aquaintance use the sort that you get given in hotels...and if you can get them film canisters are handy for chamois cream / sudofed...and if you're really stuck, Sistema make little leak-proof screw top pots (you can get 4 for about £5).
shorts weigh about 150g - so not too much burden...
If you're recce-ing the route beforehand - might you arrange yourself a mini bag drop of vital supplies?
Some "organised" 600k AUK events have facilities at the controls too.
When I had my hip done I couldn't use our over the bath shower and I found alternative facilities in strange places - including one local church.
At the risk of starting a festival of excited postings on a historically well-exercised topic, I understand mudguards are handy for keeping the bum drier than otherwise - and they help to keep road grit off that area too. Keeping the rear as dry as poss and as clean as poss is probably a good goal - because it keeps the skin in as good condition as you can do. You want to minimise the risk of getting saddle sores and anything infective getting in if you're unlucky to get one.
Do hope this helps! Enjoy your ride!