Rent a tent campsites

Glamping?

Glamping is probably as expensive as Airbnb and few places would rent you a place for just one night in peak season.

Sort of rock and a hard place here, OP. Glamping sites generally won't do one night rental, and almost definitely won't have 1 man tents, so it'll be expensive.

Normal campsites won't have tents to rent out.

Would you consider buying a light (or ultralight) backpacking tent?

Most campsites will have showers and bathrooms, have a look at pitchup to give you an idea of locations and costs, most of the time you can get a pitch for about 20-30 quid a night

That's the sort of thing I was talking about earlier, but just be aware, the model above requires a hiking pole in order to pitch it.

There are sites that offer this. Standard tents, not glamping. Two I know of are Bective Mill in Meath and Brandon Hill in Kilkenny.