Healing bomb uses a bomb and either a cure extract or a cure potion.
Fast bombs lets you throw up to you number of attacks in bombs.
Can you throw multiple healing bombs? It is potentially even better with haste.
Rules as written, I don't see anything wrong with it. It feels a bit munchkiny to have the alchemist out-heal the cleric, even if it is just for a round or two.
Is this just good resource management or do I have stinky cheese on my hands?
It's not going to be a standard tactic or a common thing, but on occasion (especially the first time you have a chance to use it), it's kind of mind blowing: Two Weapon Fighting (and/or Rapid Shot) + Haste + Iterative = 4 Cure Serious wounds in one round for 12d8 + 32 of healing while everyone adjacent to them gets 44 HP back. You can't keep it up, and you can't do it often, but the result ranges from awesome to downright silly when you do pull it off.