Box lock "My Lucks" and also a model 52, but marked with Roman numerals and the B.C. Miroku/Charles Daly model 500 are all similar SxS's w/varying degrees of finish & features. All of those models pre-date the browning connection. I have the understanding the SxS's go back to the 50's, but nothing to prove it beyond what I've been told by some Japanese collector/shooters that I know. Numbers of those SxS guns have rather short butt stocks.
I've not seen one marked exactly as you describe, but nothing from that time frame [late 50's~early 60's] would surprise me in the least. Japan was in its "coming out stage" and beginning to go global and exporting to numerous markets. Change was rampant and things like names were misunderstood and had different connotations in different markets & countries w/some translations being simply odd & not making much sense. A model F could be but a variant of an existent gun being marketed under the 'My luck' banner. I don't know.
I can't tell you much about the side lock guns except that they existed from sometime in the 50's anyway and most, if not all, of them had chopper lump bbls. and could be ordered on a semi-bespoke basis in Japan .... by the chosen few. Post war firearms laws there remain strict beyond belief. I don't know about today, but 20 years ago something as silly as modifying a choke constriction in a fixed choke shotgun was enough to get it confiscated and the Japanese owner in super hot water when the required inspection times came up. That's a very good reason to get it made like you wanted it to begin with. The two side lock guns that I have seen with the locks exposed looked very well executed and the inletting was sound as was the wood to metal fit. I've never owned one of them, so can't speak to their durability, but would expect it to be quite good.
OTOH, I've owned and used several of the BLNE's SxS's mentioned above and never had an issue with one of them. BSS's have a good reputation for a reason.
On the O/U's they started producing and exporting in the early 60's, I've broken main springs in two of the small bore small framed guns. After much use I had to have new larger diameter lower firing pins fabricated w/discs reamed accordingly and fitted in the same two guns and had the upper ejector welded back together and re-tempered on one of them. It was the earlier two-piece design & my thot was that it wasn't tempered correctly to begin with. Both of those guns have seen extended use. One went through a German importer before coming here and the other came in through Charles Daly when they were still in NY; both early 60's, Nam era. I've shot several large framed Daly Miroku 12's extensively and have never had an issue with one of them. Today's Browning Citoris have a very fine reputation and are basically today's iteration of those guns.
Perhaps something of use here for you. Best,
tw
