ventilating mildew-prone walk-in shower
You should vent to the outside. Bathroom fans are exhaust fans, why would venting to outside chill the shower? Presumably the fan unit has a flapper to prevent cold air coming in when the fan is not on. When the fan is on it will be pulling air from under the bathroom door and through the HVAC vent if there is one in the bathroom.

I think normally the bathroom exhaust fan is in the ceiling of the shower outside the shower. If it is on a timer, running the fan for 15 or 30 minutes would dry the shower enough to prevent excessive mold.

If the grout has long established mold, you might have to use bleach on the grout and then seal it.