You can get a STATT to a gun store immediately. The alternate storage application can take a little while but my only experience is with another person, not a licenced business. Mine took a month.
Just wondering, because you are taking it to a local gun store, temp storage may be less of an issue or maybe it is something that can be processed concurrently. I've taken restricteds in for repair to local gun stores and they've sat there for months waiting for a unique part etc. All I had was the STATT, but a least the CFP would have it on record that the guns were transported to the store.
Give the CFP a call and they should be able to help.