In looking through Skennertons three books (The BritishService Lee, The Lee Enfield Story, and The Lee Enfield), the typical batch size for any trials patterns was approximately 1,000 for troop testing. However, some batches were as few as 50 and as high as 2500.
Your rifle is certainly rare, but I think it a safe bet more than a two were made (but not many). In reading this section, page 187 of Skennertons last book, he makes mention of some Mk III rifles being converted to Mk V pattern. No number is given, citing the process as cost prohibitive.
This is by no means a definitive answer, nor is it even a ballpark guess. I have seen one in a friends collection, yours on this forum, Peter mentions a couple at Warminster, and Terry Hawker owns one. Counting the one in Skennertons works, that makes approximately 6(?).