Changing your number 4 days later means that the meals are not going to post on their own. To be honest, I'd say your best shot is to submit them as "missing credits" manually; I would not give iDine any explanation beyond that at that time. If they then declined to post them stating "Those dines were prior to enrolling the [new] card," I'd have to say they were within their rights to do so. However, as you say these are significant charges, I'd then appeal the denial (giving the full story) in writing. Longshot, but depending on the supervisor's mood that day, you might just get by.

Ironically, the last time I called in with a concern to my Delta account (this has not happened with any other partners), the rep insisted on verifying the details of my linked cards before proceeding to address my concern; one of them I almost never use and don't carry in my wallet (only card I have that doesn't earn miles!), but she was firm on waiting while I dug it out to confirm the expiration date.