Well, first it depends on why your GM thinks that bombs are not silent. Then work from there.
Yeah, I would homebrew a feat "Silent Bombs". If your GM isn't willing to homebrew something like that, then I'd consider getting a Wand of Silence, or consider taking the Deep Bomber archetype like Temperans suggested.
I agree with your GM that bombs aren't normally silent. Even if they're not fire, they're going to cause some sort of noise. One from the glass breaking another from the actual explosion.
However, I would allow a feat to suppress the sound, like the Silent Bomb class feature from the Deep Bomber archetype. Honestly, I wouldn't consider such a choice to be very powerful so if a player is keen to spend a feat on it, I'd say it's worth it.
Deep Bomber stacks with Vivisectionist... no matter how odd that is, it still works.
So Underground Chemist UnRogue 4/
Deep Bomber-Vivisectionist 16 would allow Silent Bombs that deal Sneak Attack damage. How fun...
Technically, it only takes Rogue 4/Alchemist 2 in order to have Silent Bombs with Sneak Attack...
At 6:
BAB +4
Base Saves +4/+7/+1
Bombs/Sneak Attack 3D6
CL 2, 1st-level extracts
Deep Bomber stacks with Vivisectionist... no matter how odd that is, it still works.
So Underground Chemist UnRogue 4/
Deep Bomber-Vivisectionist 16 would allow Silent Bombs that deal Sneak Attack damage. How fun...
It's probably an oversight, because Vivisectionist takes away bombs entirely.
But now I've got this vision of a Deep Bomber/Vivisectionist combination with a Silent But Deadly feat -- requires that you be high enough level to cast Nauseating Trail, but then also gives you Stinking Cloud and eventually even deadlier non-Elemental Cloud spells, and gives you Silent Spell with them; however, instead of using up Extracts, they key off a fraction of the number of Bombs you would have had if not for the Vivisectionist archetype (but given time and the right kind of food, you can recharge).
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Deep Bomber stacks with Vivisectionist... no matter how odd that is, it still works.
So Underground Chemist UnRogue 4/
Deep Bomber-Vivisectionist 16 would allow Silent Bombs that deal Sneak Attack damage. How fun...It's probably an oversight, because Vivisectionist takes away bombs entirely.
But now I've got this vision of a Deep Bomber/Vivisectionist combination with a Silent But Deadly feat -- requires that you be high enough level to cast Nauseating Trail, but then also gives you Stinking Cloud and eventually even deadlier non-Elemental Cloud spells, and gives you Silent Spell with them; however, instead of using up Extracts, they key off a fraction of the number of Bombs you would have had if not for the Vivisectionist archetype (but given time and the right kind of food, you can recharge).
Actually could the smoke bombs and its various upgrades be considered "Silent?"
While it does not make the bomb silent Delayed Bomb could work for a stealth based character. The character would sneak up set up a bomb and sneak away. This would also be a good way to setup a distraction. Put a delayed bomb in a place away from where you want to go and have it go off a few rounds later. When the guards rush off to investigate or are otherwise distracted you sneak by them.
Gas bomb also has some potential for stealth. Since there is no explosion, it should be less noticeable than something with an explosion. Creatures around where it was being used should still get a perception roll to notice the vial breaking, but that is going to be a lot tougher to hear than something that explodes. Since it can be combined with things like smoke bombs this may be what you are looking for.