Googling "tawas mi dog friendly hotel" brought up a bunch of links, like this one:
DogFriendly.com's City Guide: Tawas City, Michigan
Don't know if that's helpful but the first link looked worth a call....Tawas is a neat little town; my ex had a place on the lake up in Harrisville just to the north.
I have travelled and done road trips a lot with dogs for trials and shows - many, perhaps most, places accept them and it's even easier if they are smaller dogs because some chains have a weight limit. (Unfortunately mine are big...!) Usually just calling ahead and asking is your best bet.
One tip, from experience - do NOT rely only on what a website or the CS person at the 800 number for a chain tells you! I have found you can get incorrect or outdated information that way. Best to speak directly to someone at the hotel in question.