This is really more of an extended comment - my impression is that it is believed to be standard for wealthier Romans to have had some kind of bathing facility at home, but specific references or details are hard to come by.
Probably every large home had some kind of private bath, which over time became more luxurious as attested in archaeological finds.
Ermatinger, James W. The World of Ancient Rome: A Daily Life Encyclopedia. ABC-CLIO, 2015.
Note that the Romans did not acquire the habit for communal bathing until towards the end of the Republic. In earlier times, therefore, Romans either bathed in the Tiber or in crude facilities at home. For those who had it, the room for these purposes were known as the lavatrina, or washroom, and usually placed near the kitchen. Of course, this was nowhere near being universal - poorer Romans may not even have had a kitchen in their homes.
A bath or bathing- vessel, such as most persons of any consequence aiuoug the Romans possessed in their own houses . . . Thus we learn from Seneca that the ancient Romans washed their legs and arms daily and bathed their whole body once a week. The room set apart from this purpose was called lavatrina or latrina, and was placed near the kitchen, so that warm water might be easily procured.
Peck, Harry Thurston, ed. Harper's Dictionary of Classical Literature and Aantiquities. Harper, 1897.
Later writers, such as Varro or Seneca, contrasted the plainness of such rooms to the later opulence of imperial-era baths. In any case, it seems that starting from the 3rd century BC or so more luxurious bathing rooms known as balneum, began to replace the old washrooms. While still not as decadent as Imperial baths, by the end of the Republic,
[B]y the end of the Republic, as Hales and Dyson show us, private bath suites were a normal part of every aristocratic house.
Evans, Jane DeRose, ed. A Companion to the Archaeology of the Roman Republic. John Wiley & Sons, 2013.
I don't have access to the original texts so I'm unable to cite the specific evidence for the assertion. Do note by the Imperial period, everyone from slaves to Emperors used Rome's public baths, so the idea that rich people wouldn't be "pleased to encounter them at the baths" is probably not correct.