You're a helicopter pilot at a class B airport. You'd like to taxi to another parking spot. The ground controller could either give you a hover taxi clearance or an air taxi clearance, depending on the situation. Either way will get you off the ground and (technically) gets you to enter the class B airspace. Would you need to obtain a class B entry clearance prior to requesting your taxi clearance?
No. Hover taxiing with a helicopter follows the same regulations as taxiing with an aircraft (or helicopter) on wheels.
You need to be cleared into Class B airspace. Are you suggesting that a helicopter on the ground at a Class B airspace FBO is not in Class B airspace?
Technically, even with wheels on the ground anything above the surface is in class B airspace if you are at the primary airport. I have never been a helicopter PIC in class B so I cannot answer definitively, but if there is no intent to fly and you are cleared to airtaxi by ATC, (albeit a ground controller) that should be sufficient.
If you are actually facing this situation it wouldn't hurt to call on a landline ahead of time to clarify expectations with your local controllers. In any case you wouldn't need to make a separate radio call for class B entry. If you are truly concerned, this could be accomplished all at once. For example: "(callsign) request airtaxi in Bravo airspace from _ to _." Then you have it on tape.