[Half-fast eddie]

Tell the contractor he has two choices: install the cabinets to brand new condition like the contrwct calls for, or offer a price reduction. I would propose 50%, you might propose more.

 

[Half-fast eddie]

Be very careful with second contractors. First, you must give the first one the opportunity to make corrections.

 

[Half-fast eddie]

When you say 50%,

That was slightly sarcastic. You should propose a huge credit amount, enough to cover your disappointment, plus the worst case cost to make it right, plus a little more. Ideally the original contractor will fix it. If he has to remove tile, remove crown, etc … too bad. That’s the risk of being a contractor.

 

[Half-fast eddie]

I wouldn't burn bridges.

No, but if the contractor says he can’t fix it … would you accept it and look at it every day until you sell the house?

 

[Half-fast eddie]

Per OP post "He says that, because of the way it was installed and how it's attached to the soffit board behind it, ...."
I would think he knows the trouble it would cause. I would pay him his full fee and thank him for his work. As you say, the design was faulty. He had to devise a work around.
Then call a furniture refinisher. They are the expert for this job. Then be happy with your new home. None of them are perfect.
I think threatening your contractor is totally uncalled for and petty. I just wouldn't be that person. That would definitely cause me more grief than a bump on my cabinet. And don't ask contactor to pay for refinisher either. Some of the risks of homeownership has to be on the homeowner.

This from the OP indicates an oops with the screw. Installer-caused error.
“one of the screws from the wall cabinet went into the filler at an angle, pushing the wood out and creating a bump on the filler.”
The contractor needs to make it right.
“Some of the risks of homeownership has to be on the homeowner.”
Right, but the HO has insurance for risks, in addition to reserves for normal maintenance. This event was caused by the contractor, the owner should not have to pay additional to correct it. Conversely to your comment, the contractor has to assume the risk of doing business.