If Disney didn't allow them then they're probably not allowed on Cozumel.  I
believe Disney recently purchased the Puta Langosta complex which owns the
entire part of Cozumel that's safe for cruise ship passengers.  Something
about carelessly dropped regulators (like in the DIR 3 video) and suicide
clips dragged from BCs scratching up all the pretty white marble.

That would be plenty for Cozumel though you'll want to bring a cover-up for
the wickedly cold surface intervals.

I did Cozumel in Sept. '97 and Belize in June/July last year.  Water
temp was 86 F down to 110 fsw and I did it in a 3 mm shorty, no hood.
Air was typically 92 F to 95 F.  Wind can be a problem so a cover up
is good idea.  Cozumel is a National Park and preserve, no gloves, no
knives, no taking anything but pictures and memories, no touching the
reef.  The currents run 2 to 4 knots along the reefs on the west side
and all the boats practice drift diving. You don't swim much at all.
Just let the current take you past the scenery. Very nice long dives.
The dolphins came to play with us. Snorkeling off the beaches is fine,
not sure about scuba off the beaches, didn't do it then and conditions
are quite different now, I expect you'll find good facilities but the
viewing may not be as nice as it is on the six packs. The girls that
didn't have wet suits and just went in a skin or tees were cold after
30 minutes.  Drift dives lasted about an hour and the girls were very
ready to come out by then. In my opinion a full length 7mm is probably
too much. I was opening my shorty on the boat a few times to cool off.
Definitely take hats and sun screen. SPF 15 or better, probably not
more than 30 unless you are a total photophobe.  Add the hood for
insurance just in case but I never needed mine.

I would imagine the other waters to be about the same. Wear a long if
you are afraid of coral sting, but being parks they frown on bottom
crashing anyway. Buoyancy control is a must. If you lack sufficient
control, getting stung will be a nice reward.  :)

Not sure what the ships allow for gear. Consult them ASAP so you are
not surprised. I expect wetsuit, mask, fins, boots, regs are ok.  BC's
maybe; cylinders are never allowed.  I don't know about some others
but my gear all packs into a single rolling duffle/backpack except for
cylinders and weights and I've never scratched anything. I carry my
mask, regulator and camera and other essentials in a small soft
carry-on about the size of a briefcase or laptop bag suitable for air
travel. No problems with security.

If a operation did not allow me to use my own gear I would not dive with
them.  I know of no discount for using your own.