Name on health insurance card does not match name on Driver’s license

I recently changed my name after marriage and have had my drivers license changed to reflect that.

However, I just joined a new insurance plan with Anthem through my employer. I contacted Anthem who said my employer had to update my name on the insurance ID but they have been slow to do so. I have a doctor’s appointment next week and want the claim to go through properly.

Will the doctor’s office still be able to file the claim through my insurance even though the name on my drivers license doesn’t match the name on my insurance plan? The representative I spoke with at Anthem said that it shouldn’t be a problem as long as the doctor’s office files the claim under the name printed on my insurance card rather than the name on my license. Sounds easy enough but I haven’t spoke with my doctor’s office yet to know if that is allowed.

These two cards are very obviously the same person despite the different names as my maiden name was “unique first name, middle name, unique maiden name” and my married name is now “unique first name, unique maiden name, married last name” but technically the ID cards do not match exactly.