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My car was hit in 2009 and the other driver was fully liable. I told their insurance up front that I wanted to pursue diminished value and once the car was repaired they paid to have it inspected. The inspection was essentially a quick look to be sure it all seemed ok. Then the inspector, who their insurance hired, told me it was almost always 30-35% of the repair costs. Which they paid without challenging. So in this case with a $3k bill you should be looking for a $1k or so check.
The appraiser also mentioned that if the insurance company fought that claim, he'd go to court and then they'd have to pay the 30-35% and legal fees plus his time to go to court and dispute it.
YMMV, but I'd be expecting about $1k and hopefully it's just a call to the person administering your claim and possibly an appraisal/inspection. The insurance company I dealt with tried to say that to get the diminished value I had to sign off all liability for the rest of the claim including medical, I'm not sure that's legal, but it may be something they try so be wary.
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