I assume you have done a web search. Rental shops I stop at locally, just having nothing I'd want to spend several weeks on. Too heavy and toe clips. My preference would be to bring your own bike. Downside. Airline neglect and mismagement of your bike. They won't insure for their loss.
Try to find an insurance policy for air shipment? Almost each trip they have done some minor damage to my bike. I felt they were well packed.
Still, I'd bring my own. Travel on European airlines, rarely do they charge for bike transport as long as you do not exceed the allowed weight limits.