Yes, the alternator/water pump belt has to come off.Not sure I get this - you either need to remove the pulley or remove the belts on the pulley to get the timing belt onto the sprockets, don't you?
On my non-AC cars I cut the alternator belt (they’re cheap so I’m replacing it when I do the timing belt anyways), install the new timing belt, then install a new alternator belt.
But from your photo it seems you have an air conditioner? That’s a different belt arrangement that I’m not familiar with - it may be different enough that my method won’t work.