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I think it's difficult. SRAM pull ratio is so low that you need nearly twice as much cable pulled through compared to a normal shifter. From memory it needs to be able to pull about 30mm of cable to go through the whole range of the cassette. It might have enough but I seem to remember that most Shimano-compatible shifters only pull 20-25mm at most.
Not keen on TT500 bar ends? You can get a used set for £20-30
Edit: I need to mess with my derailleurs today, I can maybe try a Huret shifter on an Exact Actuation derailleur and see if can pull through the entire cassette if you like?
What are the odds of this working with a full length cable housing to a 10-speed SRAM Exact actuation derailleur?
https://www.ebay.nl/itm/SHIMANO-333-CLAMP-ON-SHIFTING-LEVER-ALUMINIUM-ALLOY-NOS/391527328771?hash=item5b28d8ec03:g:zLQAAOSwRGlXqFzq#shpCntId
This bike has no cable stops or guides of any kind, and I have everything bar shifters left over (the current levers have no working internals).
PS
I know the rear brake housing is too short. I cut it before the brake was adjusted as I was used to the arm going up as the brake is adjusted, but with single pivot brakes it turned out to be the other way round.
PPS
If anyone knows of a better solution for the bottle cage, I'm all ears. Would rivet nuts be a bad idea?