Very helpful, the new pictures reveal that the barrel is Bavarian. Note the Bavarian Lion fireproof and acceptance. It gets a bit more complicated to tell whether this is a salvaged barrel used to re-barrel this rifle or whether Amberg did the work. Amberg is known to have done some reworks and repairs (Danzig as well), typically apparently salvages (built rifles around other makers receivers).
I am tempted to be reserved in my analysis, however, because there are a few similar cases where the available information is clearer, I think I can be a little cavalier in my speculation. One rifle in particular comes to mind, owned by Wolfgang (Amberg), a DWM/05 clearly built by Amberg in late 1917 or more likely 1918. The key I think is acceptance. Making some assumptions and going by trends based primarily on known acceptance patterns, I would say there is a very good chance this re-barrel was done by Amberg late in the war. Less certainty exists because the stock can't tell us anything, in Wolfgang's case the facts are far more clear, but here there is enough similarities to go with the same conclusion.
This rifle is a little more odd in that it is a Kar.98a and not a G98, but Amberg was more than capable of doing any work necessary:
1. Amberg made the 98a 1909-1911, the acceptance on the barrel is not very clear, I really hoped it would be much clearer for a absolute match, but what is clear is it is not one of the three known acceptance stamps from that pre-war period. Further the Lion FP is inconsistent with pre-war Amberg 98a production. Only weakness on the barrel is the "OS" code is so far only seen on wartime barrels (but Imperial BC research is not as thorough as 98k), so less likely this is a leftover 1909-1911 Amberg barrel.
2. The FP and acceptance, in comparisons to 1917-1918 Amberg G98 production does have enough similarities to match. So far as trends go, the only possible match imo.
3. The few cases of Amberg wartime salvages do seem to date to this period, mostly later war, typically with C/A acceptance and these acceptance stamps look to me most closely resembling partial crown/A acceptance in Bavarian style. The FP style also fits well with this range of Amberg production. So to the suffix lower case "L" is proper for Amberg/17 style, - though obviously Amberg simply repeated the suffix on the receiver and did not serial the carbine in any meaningful way related to its own production.
4. In known Amberg builds or reworks of other makers receivers (typically salvages), Amberg typically does not apply its FP to the receiver, only to the barrel like this. (Danzig does FP the receiver), so this is consistent.
5. It is next to impossible for any other scenario to be likely. For if this was German done the fireproof would be Prussian or obviously republican, - any further German tampering would have gained another FP (probably on the receiver) beyond this Bavarian barrel, which is clearly Bavarian due to the FP and barrel code. Further the serialing of the barrel best matches Amberg with its distinctive numeric font style.
It would be much more assured if the stock was clearly marked with Amberg's acceptance, but perhaps this may have been because they didn't have a 98a stock left to use. Typically Amberg did not re-cypher or acceptance the side of the stocks like "new" production. Only the wrist and lower matches in these cases because Amberg provided the stocks.
Anyway, a great deal of "guessing and prejudice" in my "history", but I would put my conjecture more in line with reality than what passes for "science" in climate change alarmist theories...
I photographed any marking that was present on the barrel. There is some rust pitting on areas of the barrel, but I don't think that they're obscuring any other stamps.