LD1:
Although your rifle is undoubtedly in excellent condition and a very nice piece, the engraving shown in your picture does not speak very high quality.
Given the indication of 1883, I would have expected a Mortimer & Son double rifle to display very best engraving if it was a gun made, or finished, in house.
I suspect therefore that the chap from Dickson's was quite correct to say that the gun came from Scott and I have seen many guns from this source which were supplied fully engraved and ready for sale with no 'maker's number other than Scott's number on the barrel.
Only in guns made in-house, generally by the better makers, have I found the gun number on all internal parts, rarely in guns sold fully finished as this one may well have been.
Never say never, but I think you are right in saying there are no internal numbers.
Anyway, enjoy the gun for what it is!