The, "I've tried everywhere and no cats" thing is always a challenge for me.
Here's a couple on zillow - Not an ideal drive at 1 1/2 hours but it beats the expense of buying and heating an RV...
Here's one in Petoskey
For rent This 552 square foot townhome home has 1 bedrooms and 1.0 bathrooms. It is located at 1401 Crestview Dr Petoskey, Michigan.
www.zillow.com
Here's one in Big Rapids
For rent This is a 3 bedroom, 2.0 bathroom, single family home. It is located at 18500 Northland Dr Big Rapids, Michigan.
www.zillow.com
Sure there are reasons not to ask and not to try but there are options and this is only on zillow. I bet if you found some local papers or a local agent you could find a small house for example.