Theoretically, there should be a TM or TB in the 9-6625 series (OPERATOR’S, ORGANIZATIONAL, DIRECT SUPPORT, AND GENERAL SUPPORT MAINTENANCE).
Yeah that's where I started to look for it, I looked through the Library of Congress, liberatedmanuals, and the Hathi Trust library, I couldn't even find a mention of it in all the documents I looked up. I read through an index of TMs I found published in the early 80's and another in the late 90's, I didn't see it in the 9-6625 series or the 11-6625 series. There didn't even seem to be a mention of actually using it in any of the other available manuals to repair or modify equipment either. The only mention I found was oddly in a index of JETDS designators published this past June, but the index didn't provide any other information than what the AN/USM-413 was: a voltmeter.
I did however find an italian magazine on the Internet Archive that published an article about it (I assume it's a review) and other surplus equipment in 2004, it fortunately has some schematics photocopied on it, aside from that I cant really read italian that well. I also found tons of classifieds for it that advertised it with a manual, a listing in a catalog, and another magazine ad, but not much else. The article with the schematics is attached.
Which website did you buy this multimeter from?
I bought it at a local auction.