Instead of venting the air from the bath to reduce moisture why not just install a dehumidifierand drain the water into the toilet tank. Thinking about mounting wall units unobtrusively and running drain into toilet tank.
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Instead of an active cooling system, I would suggest a passive one - a reverse anti fogging mirror if you will.
Some mirrors are heated by the hot water for the shower passing behind them. Instead, you could have a radiator mounted behind a panel (or a mirror of course). The radiator would be connected with the COLD water line, so that when you used cold water to temper the hot, it would chill the coils and fins. A small bath fan would pull humid air across these fins and precipitate the humidity.
This idea is based on a similar one that uses cold ocean water to precipitate humid ocean air.
This condensed water is pretty clean BTW, you might want to save it for watering plants instead of flushing it away!
The economics of an electric dehumidifier are not particularly good. It's basically a small room air conditioner that exhausts the hot air back into the room, only dehumidifiers have not been subjected to SEER pressure like regular AC units, and hence are even less efficient. And of course they're noisy and would take a long time to clear the air.
Besides, code requires the vent fan.
What the other guys said.
Plus dehumidifiers are noisy.
We needed one while we lived in a trailer as we built the house. RVs don't usually have any vapor barrier, so it was important to control humidity during cold months. I would think in a bathroom it would be seen as big, ugly, noisy, and tempermental (you never know when it's going to kick on). My wife hated it.