This is a good question - the one thing I wouldn't do is spray water onto the bat directly - it is different for pressing the bat, you are looking at the after care of the bat.
Rather than the bathroom - I would be tempted to leave the bat in the garage - it will have less of a fluctuation of temerature and likely to have moisture form the autumn/winter air.
Definately go for some oil afterwards though. Over oiling a bat is pretty hard so you can be relatively generous with the oiling.
As alluded to above, more moisture will mean more weight.
If your bat is a GM - I believe they are treated in such a way that it is very hard for them to gain/lose moisture.