Laurie - Would you leave well enough alone if you got a quieter fan? They do make them. Usually, fans make a lot of noise when the door is close because they're 'fighting' to draw air to 'push' outside. You don't need much to 'vent' the room, hence, you don't need a 'fast, noisy' fan to do the job.
Measure the cubic footage of your bathroom (L x W x H). Bring this number w/you to the Home Center. The fan box will tell you how much air in CF (Cubic Feet) the fan will push in a minute. If your bathroom is 360 CF, and the fan is sayin 150 CF/Minute, then essentially, the fan can 'clear' the bathroom in just under 2.5 minutes. (150 x 2.5 = 375) Talk to the Rep and get his 'thoughts'.
My best to ya and hope this helps.
Jay J -Moderator
PS: My reply doesn't help you with the wiring but I thought it to be another 'option'.