Rapid Shot is an extra attack. At level 3, your BAB gives you 1 attack. Therefore, Using Rapid Shot during a Full Attack Action allows you to make 2 attacks at your highest BAB-2.
Using a double barrel weapon and firing both barrels in the same round on your turn is a special standard action attack.
This musket has two parallel barrels; each barrel can be shot independently as separate attacks, or both can be fired at once as a standard action (the attack action).
If both barrels are fired at once, they must both target the same creature or object, and the gun becomes wildly inaccurate, taking a –4 penalty on each shot. Each barrel of a double-barreled musket uses either a bullet and a single dose of black powder or an alchemical cartridge as ammunition.
Emphasis mine.
I'm not sure what Lightning Musket is...
But I will guess you mean Fast Musket, which if it is what you mean, then no, a level 3 musket master with rapid reload reloading a firearm with alchemical cartridges does not cost a grit point, but you must have a grit point in your pool in order to do so.