This is another example of how the acquisition process and registration process get co-mingled and confused.
The PERMIT TO ACQUIRE has one purpose: to allow the transferee to legally acquire a firearm. In the case of a long gun, the permit TO ACQUIRE has a 12 month expiration date from time of issue. If you use it to acquire a long gun before the permit's expiration date (proven by the seller notifying HPD of the transfer as well as a dated receipt or bill of sale), then the transfer was legally completed even if the permit expires the day after the transfer.
The REGISTRATION process is a separate requirement from the transfer process. They want you to show the expired permit you used, because that permit is tied to the transfer transaction.
If I were in the situation described, I would make a copy of the expired permit to take to my registration appointment, then book an appointment to renew the permit. If the renewal happens before the registration, then I have the copy to use.
A copy of an expired permit should be just as valid as an original expired permit for registration purposes.
I think as long as they get the permit number and expiration date from a copy of your expired permit, they should be satisfied.
This is my opinion. Best to fly this by HPD in advance if you intend to try it.