Got it! and good catch!
I never put the 2 and 2 together as for that little grub screw in the breech face.
I am often quite conservative in taking a gun apart, especially an unknown mechanism... But I'll certainly go for it now.
The system seems to be a fairly standard Anson with removable access plates. The sears appear to be on the back of the tumbler (overhanging) instead of the traditional bottom.
It reminds me of my Francotte FASD ("Francotte Anson Systeme demontable" which is an A&D mounted on sidelocks without bridles.

It seems that you are hitting difficulties with the regulation. I am worried that the center to center distances of your gun are identical to mine, while it has shorter barrels. That would not make sense (at least to me).

As for the rest, I am now totally convinced that the 1897 is the manufacturing date, and that the 35xxx number on the rib was imprinted by Bernard.

Best regards,
WC-