In order to utilize a F-10, the museum needs to be run by the government. Any internet/gun shop commentary on funding sources, etc. are in error. See 27 CFR 479.104 ("governmental entities").
Same for the captured enemy equipment certificates. Commonly cited on the web as defacto registration, but to date I have not been able to get anyone claiming that to cite a verifiable example and the process is not in law or regulation anywhere. Also inquired with two real life gun lawyers (Barnes & Halbrook) who at that time had not seen any such examples. I have never seen anything that connected War Dept forms issued in overseas regions with Treasury registration in the US. This myth may be somewhat rooted in truth due to an E TN appeals court case in the 70s (McKeehan) where firearms brought back and subsequently transferred paperless were protected from confiscation. That concept is not being applied to anyone else.