Just for grins,,try backing off on the rear tang screw first. A 1/2 turn. You will need to take the TG screw out and unscrew the TG out of the way to get at the rear tang screw.
Then, back off the top tang screw (underneath the Top Lever). Same thing,,1/2 turn.
What this does is relaxes the trigger plate a bit and can actually allow it to straighten some if it was bowed a bit by the tightening of the screws.
This can all allow just a few .000 more room betw the trigger mechanism and the sear arms.
That can let the Trigger Selector move as it should to positions beneath either of the sear arms when selected.
If the selector in the Trigger Mechanism can't completely move side to side , it can be halted in a position where it lifts both sear arms at the same time
no matter which side is selected.,and the gun doubles.
Just a thought to try, It doesn't take much work, remove TG screw and unscrew the TG a bit,,then back off the rear tang screw. Then back off the Top Tang Screw.
Then try the gun and see if the trigger will select properly and fire the selected bbl w/o doubling. Then will it also continue to work and fire the other bbl as normal.
Sometimes those 2 screws can be backed off a bit more and sometimes less to 'make it work'.
The Glass bedding of the action and trigger plate can sometimes affect the trigger position if it's not carefully restricted to keeping the upper and lower tangs separated an exact amount.
Just some thoughts. Single Triggers can be a tough thing to solve. A lot of 'smiths won't even work on them.