I once had to have a rat euthanized by the vet (and I was extremely fond of
this rat!).  The vet, however, refused to let me see or hold him during the
process, as they inject him in the stomach with an extremely large needle.
Vet had previously found this was too distressing to the animal's owners.
 
If the ferret must be put down, the swiftness of a pellet gun (a ferret
wouldn't even know what the object was!), versus a large painful needle
injection (and I'm sure he won't understand why you are putting him through
more pain in a stranger's hands) is not too much of a difference.  Dead is
dead.
 
I think the best decision is the one which makes you personally feel
better.  The pellet gun causes a vivid injury which touches your primal
brain.  It is brutally unsettling.  Yet if you view it
logically/unemotionally, it is so quick as to be entirely painless.
 
P.S.  After saying all this, if it ever came down for my ferrets, I'd
probably take them to my trusted and known vet, rather than to a stranger,
even though that stranger may be very gentle and experienced.
[Posted in FML issue 2571]