You will gain more by upgrading your rear suspension and riding without the backrest and you may find it solves the problem and allows you to sit further back than the 6 position adjustment wheel on the Rider's backrest allows. You can then sell the backrest for a small fortune and buy a Utopia one that fits the same hole in your seat for half and I believe they have much more adjustment available! 
I only have a couple of the R3 Standard/Roadster 2 piece seat version which bolt on differently but the Classic and R3Touring backrest version for the dual seats will also fit the R3Touring Solo seat.
https://www.utpr.com/rocket.html
I gave the top bidder a tickle and he has increased his bid. I have seen them hit US$350 or more.
Triumph Thunderbird 1600 1700 Storm Rocket III Touring Rider Backrest Longhaul | eBay
