If the hotel shuttle service insists they cannot take your husband to the International terminal, then it would indeed be less hassle to take a cab from the hotel directly to the International terminal ("F"). Otherwise, if the shuttle were to drop him off at the Domestic terminal, he would have to take yet another (infuriatingly slow) shuttle that runs between the Domestic and International terminals.
All nonstop flights departing from Atlanta for international destinations (that is, with no intervening domestic connection) depart from the International terminal. Passengers on these flights are not permitted to check in at the Domestic terminal (hence the free shuttle between Domestic and International). Rather, they must check in by entering the International terminal directly from the outside.
You might logically think that because there is an underground train that ferries passengers among various gates along the entire route (over a mile) between the Domestic and International terminals that they would simply let you arrive at the airport at either end--Domestic or International--and then board the train and take it all the way to the opposite end, but that is not the case.
The system especially sucks for us Atlantans. The MARTA light rail system ("subway" or whatever) that I can take from where I live to the airport serves only Domestic and not International. So when I leave for Bonaire via the MARTA, I arrive at Domestic and then immediately walk outside and take the shuttle to International. Ditto on the return trip. On the return, when I land at International and exit Immigration/Customs, I am not permitted to board the train that could very easily take me to Domestic where I could catch the MARTA. Rather, since I don't have an onward boarding pass, they wouldn't let me through. So I need to go outside the International terminal and take the infuriatingly slow shuttle bus back to the Domestic terminal.
I hope this clears it up for y'all!