No, HMRC will not give you this info.
I don't think there is a sure way. The critical thing is the date on the invoice - they should have back-dated it to the date of the original transaction - that is the tax point date.
If they have not done that, demand that they re-issue the invoice to the correct date. If they refuse to do that, assert that you believe that they were not registered for VAT at the tax point date, and that you were therefore charged VAT incorrectly. See if they back down. They might think that they could fall foul of HMRC.