Thanks, Paraxis! Good thinking. That may work as well.
Let's change the situation a bit...
The first room is empty (no bandits), but the door to the second room (with bandits) is locked and the party's rogue fails to pick the lock.
The party proceeds to bash down the door, alerting the bandits in the next room.
There wouldn't be a roll for initiative before the door is bashed down, and neither side would be surprised; the bandits hear the party bashing at the door and the party is preparing for whatever lies on the other side. In this case, I think MerricB's suggestion of using the surprise mechanic in order to give the bandits the appropriate edge over the party makes sense here too.
Any thoughts?
In this case I wouldn't use surprise for the heroes, possible surprise for the bandits if they hide in the room and wait to ambush them if they have time to do so would depend on the DM and how long it took to pick the lock and break down the door.
But if neither side tries to hide or act with stealth there is no surprise in 5th edition.
I can see a DM imposing disadvantage sometimes, or grant advantage sometimes, to initiative checks it all depends.