I looked through my trusty copy of Modern Calendar Watches printed in 1954. Your module looks like the diagram attached. It's a 'Record Datofix' calendar, no model number is given. It appears the lever 'L' has the tip broken off making adjustment of your moonphase impossible although it should still work while the watch is in operation, it is just not adjustable to be in sync with the proper moonphase. I've also attached the technical notes on this movement.
Hope this helps.
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