Is it possible to dispute a hard inquiry?

A hard inquiry can drop your credit score between 5 and 10 points.

You can dispute the hard inquiry either with the creditor or with the credit reporting agency. Since your creditor gave you a verbal assurance, so you can ask the creditor to remove it from your credit report.

Do you have papers to prove that the card was pre-approved?

If there's a hard inquiry in your credit report, check out total how many are there. Know that several queries over a 45-day period is considered to be 1 hard inquiry.
Since it is a verbal pre-approval, you don't have any document to prove it. Only you can request your creditor to drop it from your credit report. Your creditor may do it due to goodwill.

Do you have any document regarding the pre-approved credit card? If you don't, then you can't dispute it with the credit bureaus. There's no other way except to contact the creditor and request him to remove the hard inquiry from your credit report. Do it asap.. unless this hard inquiry may remain in your credit report for long 2 years and surely damage your credit score by 5 to 10 points.