Hm, good point. We could set aside a few instants for him to send a few photons that wouldn't depend on the state of the particle. From a practical standpoint that's pretty impossible, but forget practicality.

So, sure; Bob should accept the bet. Although if he makes his answer to the bet depend on the state of the particle at all, then he shouldn't accept the bet :P There might be some interesting applications of this, say where the option "don't accept" has some positive payoff if the particle changes directions. Bob can precommit to send out an entangled qubit to get some chance at that reward.