You may want to try to contact golden triangle bike rentals (bikepittsburgh.com) or bicycle heaven bike museum (bicycleheaven.org), both which offer rentals and inquire about a monthly deal.
You may also be able to find a deal on a used bike at thick bikes and ride it till you don’t need it and try to do a quick sale before you leave town. Someone on the forums may be interested in doing a bike-rent deal depending on your size and bike needs. Could also try spinlister (like airbnb, for bike rentals), but I have no experience with this.
My guess is that it would be cheaper to buy a used bike than to rent one for a few months.
In Hawaii, there was a place that would seel you a used bike and them buy it back (for somehting like 20% of the sale price) when you left. It didn’t make sense for the week that I was there, but would be a great deal for a longer stay. The main motivation may have been insurance costs.
Maybe one of the bike shops here would make a deal like that.
How big are you? Someone here might loan you a bike.