having spent many years playing chord harmonica in a professional trio,
i have to say i've never seen any of the good chromatic players ever
have to clean the reed plates of their harmonicas.

there is some required maintenance on the slide plates in the mouthpiece
of the instrument, which can be cleaned with a light touch of sos or
brillo pads.  be careful to completely rinse the parts, dry thoroughly,
and use a very light silicon oil on the slide itself.  just wetting the
end of your forefinger, rubbing it against your thumb, then running the
slide between your thumb and forefinger is sufficient.  There are two
rubber grommets, one on each of the two mouthpiece mounting screws.  
Make sure to put those back or replace them with each cleaning.

as for not gumming up the reed plates, all the pro's i've ever talked to
have learned to "play dry", meaning don't eat before or during a
session, keep your head fairly upright most of the time, swallow when
your mouth feels wet, etc.

my chord harmonicas run about $1,600 each, and in the 30 years i've
played, i've owned 6 of them.   the reason for that is because i tend to
really rap some of the chords and the reeds break after extended abuse.  
they aren't the same reeds as used in chromatic harps, so they're hard
to replace.  i've never had a chord gum up on me.  my lead player, now
deceased, played his favorite 64 chromatic for well over 30 years and
never once had a problem with gumming up the reeds.

if you really really REALLY have to do the efferdent thing on your
harps, you might look at using the hohner CBH harp.  its made mostly of
delrin i think, no wooden parts.  the liquid dip is almost certainly
going to fry the valves, but i don't see any other solution.

i really suggest learning to play dry.

hope this helps

karen