carry a bicycle in a Mooney?

I have an M20J and I'd like to be able to take a bicycle sometimes. It's
OK if it doesn't fit through the baggage door, just needs to fit
inside the cabin and not block access through the (only) door. No
passengers, just me in the left seat. I'd prefer not to remove any
seats. I'm willing to accept compromised riding qualities to get one
that fits. Has anyone done this with a Mooney? What bicycle brands and
models will fit, if any? Thanks.

DB