iserith
Magic Wordsmith
I don't agree.
Once you've taken the "hide action" you can maintain your stealthiness by moving at no more than half your speed. That is, for lack of a better term, sneaking. You can continue to make your stealth rolls as you sneak around at half speed without having to take another "hide action" until you are discovered.
I think that an invisible character should absolutely be able to sneak. If the character uses its full movement, then sure--it is making no effort, or no meaningful effort--to be stealthy. If it uses no more than half of its movement and the player expresses an intent to tiptoe quietly, then it is entirely reasonable to let that character make a stealth check as he sneaks across the room.
I don't think that ruling is unreasonable and as a player I'd gratefully accept it for the gift that it is. As a DM, I don't think that ruling is in keeping with the rules for hiding and combat action economy and thus my ruling would differ.