"Blitzlichthalter" ! 😀
What you have there is the "official" Leitz CEYOO flashlight holder.
These came-out in the 1940's, and lingered into the M era.
There were a couple variations, some with a 22.5 volt battery-capacitor pack, others that used three or four "AA" penlite batteries( don't remember which).
There was also the CTOOM flashlight bracket which attached to the base-plate of the camera, and had a mounting foot for the flash that swiveled radially (in the same plane as the film).
The flash gun itself came with an adapter to use the Bayonet-base bulbs, such as the # 5, # 25, and SM, and if this was removed, revealed a larger socket that would accept Edison medium screw-base "Press" flashbulbs. Additionally, third-party manufacturers made adapter sockets that were essentially a bayonet-base that snapped-into the socket, and allowed the photographer to use the M-2 and M-3 bulbs.
If you wanted to pursue bulb substitution to the very end, I think there was another adapter to take bayonet-base or M-2/M-3 base to the AG-1 "peanut" bulbs.
The CEYOO flash holders all used the Leitz proprietary two-prong plug to connect to the flashgun, and then either another two-prong plug to connect to the VACU clip-on synchronizer, or the standard 3mm Co-axial PC plug for the backside of the -f and -g Barnacks, or the proprietary Letiz co-axial plug for the M-3.
If the adjustable flash sync on your Barnack is working correctly, and you want to shoot flash pictures, an electronic unit is the cheapest way to go, in the long-run... flash bulbs are getting expensive.
Vintage electronic flashes can be found - the Heiland/Honeywell Strobanar "potato-masher" flashes are common and fairly reliable.
If you simply must shoot bulbs ( and I understand the urge to do so... 😉 ), the bayonet-base M, SM, 5 and 25 bulbs are "classic", and the M-2 / M-3 bulbs are probably cheapest and will work well. If you're shooting color films, use blue-coated bulbs.
Good luck with your project !
(PS: there was also a smaller flash-attachment specifically designed for the -f Barnacks: the white-plastic CHICO.)