I haven't done much of Route 66 but I've done quite a few miles on bikes and in cars in the US.
I've rented cars and bikes and car hire is MUCH cheaper than bike hire.
In May I rented a brand new Chevy Malibu for two weeks (through Hertz so no small back street place) for virtually the price of a bike for 2 days.
If you went for a small car it would be even cheaper.
Don't forget though a bigger car might be more suited to the miles you're planning and petrol is cheap (cheaper than the UK anyway).
Deposits are cheaper too. Bikes are, I think, min $1000 whereas I think my car was $500.
Don't forget to tell your credit card people you're using it abroad or they might block it. Ask me how I know
Enjoy your trip
