You can clean those carbs without removing them, although it will be a bit tight. But it is doable as I clean mine all the time without totally removing them.
On the SXR, do the following:
1. Remove the airbox
2. Remove the three fuel lines that go to each carb from the fuel pump
3. Loosen the clamps that hold the carbs to the intake manifold
4. On the chain case side and the clutch side is a clasp that holds the coolant
hose; bend each one up to allow the hoses to extend a bit (this is needed to
give additional room for working on the carbs). At this point, you can stand the
carbs on end; the clutch side up in the air and the chaincase side down.
5. Put a rag down underneath the carbs. Also, have a small container handy (a
paint can top works great).
6. Remove the bottom plug of each carb (17mm). While unscrewing it, place the
container underneath to catch the fuel that drains out of the float bowl.
7. Remove the four pan head screws that hold the float bowl. Don't loose the
large, rubber gasket between the carb and the float bowl and be careful of the
moving parts on the floats (don't bend anything).
8. To keep things easy, simply spray carb cleaner in the jets/passages. You can
remove them, but this method should be sufficient for an "on engine" cleaning.
9. Put the float bowl back on (be sure to place the gasket in the right place and
don't pinch it.
10. Remove the main jet; this is the one visible when you removed the bottom
plug. You didn't remove it when the float bowl was off because the retainer
could easily get lost. Clean it with carb cleaner and check it's size. (To
remove this, take a short flat head screw driver and get it loose. Then take a
tooth pick, gently shove it in the jet and unscrew it with the tooth pick. To
replace, repeat the process in reverse).
11. Replace the bottom plug for the carb.
12. Verify the air screw setting. On this carb it's found on the lower front towards
the intake manifold. It's a small, flat head screw. Turn it all the way in until
it lightly seats (not too tight). Then back it out 1 1/2 turns.
13. Repeat the above for the other two carbs. The third one get's kinda tricky
since you're not working with a lot of room.
14. Replace the carbs, tighten the clamps, reclasp the hoses, replace the fuel
lines, put the airbox back on, be sure to secure those clamps and make sure
there are not leaks from the airbox to the carbs.
15. Double check the cables to the carbs as you may have pulled the ends out of
the cables where they connect to the carbs. Also, route them correctly once
the airbox is back on.
16. Because the float bowls were drained, it may take 15 pulls or so to get 'er
primed and started.
17. Hopefully all will run well.
Cheers,
Jon