@Raudhbjorn I can't really answer you in any relevant manner without spoiling future chapters, concerning their circumstances in particular.
Generally speaking, however, it would be a massive bother to go to some public bath every time you wanted a shower. How cheap could they be anyway to be called cheap seeing how often a sanitary person needs to shower? Maybe if they had a system like for 2,500 yen you can visit as many times a month as you want. That would be quite excellent, especially if there was a sauna and everything. Kind of doubt it.
as an autistic person this is refreshing to read. her portrayal is a little stereotyped (definately fitting the aspergers stereotype but that diagnosis doesnt exist anymore) but it's still pretty decent, theres also not a lot of female autistic representation out there so its good to see some.