Hello and welcome to the forum. The fact that you already have an A-type overdrive on a TR6 transmission is proof enough that the transmission can accept this type of electric overdrive. The J-type was the norm for this car in the later years but the A-type certainly will fit and work, and is preferred by some because of it's ability to shift harder and faster. Did you do the work on the transmission yourself or did you source it out? Any non-overdrive Triumph 4-speed will have to be modified to accept either overdrive. Contact John Esposito at quantumechanics.com for questions about the conversion and the pricing of either the kits (which he sells for the do-it-yourselfer) or the cost of his doing the conversion. John is one of the best in this area. Again, doable, will require some shaft conversion and other parts. Don't know if you will or can do the work yourself or ask someone like John to do it for you. Doubt that John would just do the conversion without insisting on reworking the overdrive unit in order to warrantee his work but I could be wrong. Too, I believe that there is a shop in your area that work with Triumph trannies. Someone on the forum will pipe up and attest to this. Good luck.
Point of interest, A-type overdrive units are getting more and more difficult to find, and valuable.
Bill