I’d go into town and try your luck, knowing you may have to pay the difference out of your own pocket. Enjoyment factor will be much higher when you’re already annoyed about delay.

LH will take WEEKS to respond to your claim, so be patient.

If you had status with LH they’d be more willing to help. Last week they got back to me paying my suite at a 5* hotel and airport limousine transfers in Germany after a weather related overnight misconnect in Frankfurt. I was on a First class ticket though and have status, but in any case, I found the result very satisfactory - no pushback, no argument, just a response apologising for the delay and saying the money has been sent to my account..!