My wife and I did a lot of shopping for a lender for our mortgage. The strategy we used, the first lender who ran our credit was required to share it with us. We took that copy to all the other lenders and had them run the numbers assuming that the number was the one in hand. Since it's a recent report, they were able to tell us the rate, closing costs, etc. without running it again. Obviously, the lender you choose will have to run it again to verify. We didn't get dinged for inquiring with 7 or 8 lenders (it was 7 years ago). Expect some weird looks and some push-back for this strategy as the lender will want to run it anyway "to make sure they give you an accurate quote". With so many estimates, we were able to negotiate closing costs and rate. We got 0.125% off and closing costs down to $500. The "damage" for inquiries is usually small unless it gets excessive.
If you haven't received your free credit report for a year, you could try taking that with you too.