Whatever the rifle is, I can guarantee you it is not 223. That drilling, built on the Roux, underlever action, is probably pre-WWI. The daddy of the 223's designer was a toddler, if he was even born when that drilling was made. That and the case head the 223 was derived from didn't appear until 1950. I'm not sure what is going on with the shotgun barrels but to have 12 and 16 in the same barrel set is highly irregular for that period of time. 20 bore is equally unlikely. But, who knows what was done to it between the time it was made and when Grandpa put it away. It's equally possible someone had it built that way for themselves....but again, highly unlikely.
Firpo is correct. You're going to have to take it down and post pictures of the proof marks and they are where he described. Take down is easy. On the bottom of the forearm should be a release. With the action closed, pull it and the fore arm should slip off into your hand. Set it aside, then holding the barrels, lower the operating lever and the barrels will swing down and can be lifted from the receiver. If possible I'd like to see pictures of at least a portion of the shotgun bores. Is there any rifling present in either one, straight or helical?
I don't know where or even how anyone determined there was only 3 made. I've owned more than that and have 3 in the toy box as I type this.....and there is probably hundreds for sale on line right now, not counting those still tucked away in closets, attics, garages and who knows where. It's impossible to know how many were made, even in that configuration, if it's correct and original. Most were brought to this country as war trophy's.
S/N's weren't required then and except for the very largest makers were not used.
Belgian is possible but German is one heck of a lot more likely. The Tyrolean cheek rest screams Germany as well as pre-WWI, same as the Roux action. The proof marks will tell the story. Date of manufacture can only be narrowed to a range and again, need to see the proofs to tell. Without seeing them, everything is speculation.