journeymen,
Gauge number 29.43 is way too large for that type of action. However, 129.43 would be a good fit. It translates to a bore (not groove or bullet) of 8.38mm. This is about .303" and would be appropriate for several of the nominal 8mm Tesching, 320, or 32 caliber cartridges, many of which could be used in this type of action. It is a pre 1912 gun and since it has been relined the original chamber is not available for making a chamber cast. Consequently, I'm afraid it is no longer possible to determine the original chambering. It is my unsolicited opinion that modern 22 Hornet ammunition would generate too much pressure for the long-term use of the gun. I suggest you load it to 5.6x35R Vierling pressure instead.
Mike