File the claim.

Generally, your insurance company can only raise your premiums for traffic violations, accidents or damage that are your fault, adding drivers to your policy, moving to a higher risk locale, or letting your insurance lapse. There may be other reasons applicable to your situation, but probably not.

Why pay insurance premiums over and over if you're not going to file a claim when you finally have a covered event? Now you will be out all those premiums, plus the interest you could have earned over time on those premium dollars had you kept them, plus $5,000 to $6,000.