No way to fit a Flues SxS to a SBT, and I agree with Stan the buttstock is a replacement. The wood pattern is not nearly nice enough for a Flues SBT.
The FE wood may be original, but the pushbutton escutcheon should be case colored. One of the issues with the Flues FE wood is cracking around the escutcheon.
I also suspect the frame has been (tastefully) torch colored.
The other issue with the Flues is the cocking lever breaking.
Some very nice Flues here
https://www.trapshooters.com/threads...lineup.894455/
The Flues bolting evolution
https://www.trapshooters.com/threads...hanism.861469/