Yes, it is necessary to open up the bellows to get to the inner flap
valves on a Gem Roller Organ.  (I understand that it's "possible" to do
a "bellows-ectomy" and surgically operate on the flap valves, but that
would only make sense if the bellows cloth had already been replaced
with new.)  Those who have commented are correct: the best solution is
simply to replace both flap valves and all bellows cloth at the same
time.

Also, about matching new screws to old:  Since there are only a few
screws on a Gem, and since modern replacements are the same shape and
size, if some are missing I usually just replace them all.  (Of course
the original screws are added to my parts bin for the next job.)  If
screws are missing because the wood has been stripped out, glue in
a small wooden plug before reinserting another screw.

Todd Augsburger - Roller Organs
http://www.rollerorgans.com/