That is incorrect information, your FEO is mistaken.
A firearm which is being acquired for the “good reason” of humane dispatch would need added as such but a rifle obtained for deer stalking or vermin control does not need to have humane dispatch listed to be used for this lawful purpose.
See page 106 section 13.9
For example, you are out lamping rabbits in Scotland with a .17HMR on ground where you have permission to shoot. The .17HMR is listed in the conditions on your certificate for the shooting of all lawful quarry.
Whilst spotlighting you observe a deer stuck with a rear leg in a stock fence.
It becomes obvious that the deer has catastrophic injuries and in order to end its suffering, you humanely dispatch it with the .17HMR.
This is completely acceptable under the provision of section 25 Deer (Scotland) Act 1996 and there is NO requirement to have humane dispatch on your certificate to use the rifle for this lawful purpose in the circumstances described.