You can still use the heat mat, but you do need to put it on a thermostat - one of these would be fine:
http://www.camzoo.co.uk/pd-1167048544.htm?categoryId=12
Would also be a good idea to cover it with something like a large ceramic tile so that the snake can't come into contact with it, and put some insulating tape around the terminal block (the bit of the mat where the wire attaches) to help prevent any moisture (eg. runny poo) getting in there. make sure it's well stuck down though, otherwise the snake can get stuck to it.
The thermostat has a probe which senses the temp and tells the stat when to switch on and off to keep the temp stable - the probe would sit on top of the tile, or on the surface of the mat if no tile is used.
You'd need a decent mercury or digital thermometer to set the thermostat - you can't rely on the figures on the stats dial, they're rarely, if ever accurate.
If you set the temp over the mat to around 86F/30C this should be fine for him.
Hopefully he's got a couple of hides in there already - put the warm end hide directly over the mat so that he's got one cool hide and one warm.
As for bathing him - you could do it in the bath (rinse it thoroughly before and clean it afterwards) but I prefer to use a large plastic storage box. Most of mine hate having a bath but with the plastic storage box I can just secure the lid to stop them diving out all the time.