Finding bike for hire in Bangalore/Mysore may be a little dicey... they are not the usual places where people hire bikes  There are quite a few places in Goa, but Bonny Sonny Motors come recommended.
One option could to be hire a bike in Goa. You could ride down the coast to Kerala (should take a day or two to do this, be warned there are plenty of places to see on this road) Alternately, you can transport the bikes by train.
Suppose you end your trip at Chennai (hypothetically) then you can use a road transport corp. (i.e. truck) to get the bike back to Goa.
Hope this helps!
Sushil
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