It might be too small for an HVAC person to take it on. It's somewhat hard to recommend because it does involve wiring, and perhaps some siding and flashing work. Many locales require electrical work to be done by a licensed electrician, and you don't want some bozo not doing the flashing correctly if they mess with your siding.

You should be able to run the exhaust duct through the joist bays that make up the bathroom ceiling and through the rim of the cantilever. It could exhaust downwards out of the cantilever soffit, but that can sometimes not be the best choice.

A small remodeling outfit might be well-versed all aspects of what you need to have done. A capable and conscientious handyman? That can be good or bad, depending on the person.

Consider asking for, or use these as points within the conversation:
-Rigid duct. Nothing "flex" or with ridges/corrugations. They can use a small section of flex to connect the fan housing to the run of rigid, but there should be no tight radii in the flex or any kinks in the flex.
-All seams in the duct sealed with aluminum foil tape. Not "duct tape".
-Insulation around the duct, at least within 4' of the rim.
-Any horizontal runs in the duct will be pitched down for drainage in case the moist air produces condensate within the duct.
-Ask how they will flash the damper into the siding of they run it through the rim.
-Ask how they will damp the exhaust if they run it downwards through the cantilever soffit.
-Consider getting a timer switch for the fan. You turn the shower light ON, the shower light and exhaust fan turn on. You finish showering and turn the shower light OFF, the light turns off immediately, but the fan continues to run for a time set by you; another 20 or 30 minutes.

Do get an idea of how many feet of duct you'll need. Fans are rated to move a certain amount of air, the longer the run of duct and the more turns within that run, the more resistance to flow and the less air the fan can actually move. You might need to up-size the fan to actually move the needed amount of air.

Wow that's all very helpful and reaffirms that I do need someone competent to do this. Sadly my capable and conscientious handyman is so busy he doesn't have time for me anymore. I can understand though because good ones are hard to come by. I also have a good small remodeling builder/contractor and maybe if I combined this with some other work I wouldn't feel like I was insulting him with such a small job.