The letter of authenticity request is the one you need to do first. The second form is for a search of the SWHF archives for additional information.
whether or no you request the search will depend on what you find out in your letter of authenticity. The SWHF indexed archives primarily cover the 1930's but is expanding. If your guns is fro the teens or the 50's for example, they are not likely to have any additional info.

Good luck on your research.

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Randy
Provenance nerd