Nice 17th badges Andy
I have included a picture of Private Eric Bagnall, 17th reinforcements A Company, Service No. 26762.
In Regards to an "Registered Design number" your badge frame does not have one, it is a generic design that was first worn by the 13th Reinforcements, and was primarily sold by Trentham Camp jeweller Horace William Lloyd.
17th reinforcements A Company, Private Eric Bagnall, Reg No 26762.jpg