The reference to "Royal Garrison ARTILLERY" in my initial reply was an error; the extract from "Boer War Services of Military Officers" SHOULD have read "Lt., Royal Garrison REGIMENT". Brain snap - sorry about that. Four Bns. of the Royal Garrison Regiment were sent to SA in 1904 for simple garrison duty; training, etc. As the conflict ended in 5/02, no medal for that. The RGR was disbanded in 1908.
On the WO100/193 medal roll for the 2nd Northamptons, on page 2 I saw "2nd Lt. A.G.L. Pepys" as entitled to clasps Cape Colony. Orange Free State and Transvaal on his QSA medal.
If you are certain that your man was present on the Jameson Raid and subsequently served in Rhodesia, you need to somehow make the link. I would have thought that a man who saw through four campaigns would want to see his diverse service reflected in official records.
Keep us posted about your discoveries.