"Balneō lavor" should be added.
Or in balneo lavor?
Sure. In + ablative case, or just the ablative case, to mark location. Unless it's one of the words that still have the locative case in classicsl Latin, in which case that is used to mark location.
Pardon me, but shouldn't it be "se" lavo, since the pronoun is directed at the subject of the action?
No. It's "me lavo".
The same way you say "I wash myself (me), and not I wash oneself (se), or I wash her/him.
The reflexive agrees with the person it "reflects".
I (me/myself), You (you/yourself), etc...
I have tried the same words in another order, but was rejected with: In balneo lavo me - and with: Me in balneo lavo. Why is only one word order allowed in this sentence?