Dear Mike Ford,
you sound a bit disappointed. And so was I. My gun is conspicuously devoid of any ciphers, trademarks or texts that could be indicative of the actual (whole or parts) producer. The overall design is what we sometimes find called a "Suhler Einheitsdrillling", although the moniker is a bit misleading ("Einheits....", in German, connotates a given and imposed standard). Also, its locking system (two wedges into the respective barel underlugs, activated with a underlever) is widely called "Roux" in Germany, but old Henri Roux had almost nothing to do with it. I have just learned that the patent for this configuration allegedly went to Purdey already around 1863.
However, you have already indicated that the hint to the manufacturer (or parts deliverer) might frequently be some minute details in the configuration of specific parts, because parts and almost-finished guns were frequently made for the trade by only a few big players. ESHA (Schmidt & Habermann) was one of them.
Now let's look at the other Big Cat, namely at Jacob Römer's "Römerwerk[e]". He changed the spelling of his last name and became Röhmer (with the Röhmerwerk) at some point. The company details have been tentatively (but not really in-depth archivally) researched and hav ebeen partially documented in the "Germanhuntingguns" website, founded by Dieter Apel, rebuilt and mainted by Larry Schuknecht.
And indeed, a quick net.search just provided found pictures of one gun with this selfsame very peculiar escutcheon on the right side of the stock near the grip, in its characteristic claw-like or beak-like configuration. Maybe this is the "Eureka" moment ? ;-)
Here is the link to some pictures (the asking price is about one thousand € too high, by German standards, but that is typical for Buchmiller):
https://www.progun.de/haendlershop~gebu-waffenhandel_georg_buchmiller/filter/ac9195x6_roemerwerke_suhl_drilling_graviert_-_16-65_u_8x57ir.html
Regards,
Carcano