There is a second fuse, but this goes to the battery charging system
and seems to be unaffected by the stopping sequence. �
Thanks in advance.

Fuses have other characteristics than Amps. What type does the owners
manual recommend?

GSS wrote:
I have a gas powered 6.5 KW Kohler generator Model 6.5CZ that blows
the 10 amp input fuse whenever I shut it down from the engine mounted
or remote kill switches.

This may not be related. However, I have a 20kw Kohler that kept blowing
fuses for no reason. I found out the improper fuse was installed. A 10
amp high heat ceramic fuse should have been used. The heat given off
from the generator engine would heat up the fuse, and de-rate it. A
ceramic fuse is designed to operate in high ambient temperatures.

I hope this helps.
John