A quick and clean way to bleed brakes on a snowmobile is to slide a hose over the bleed screw on the caliper and route the other end into a container. When you loosen the screw, the old brake fluid will flow out of the hose and into the container rather than dripping behind the chaincase and inside the nosepan or bulkhead.
As you pump the brake lever to flush the fluid, keep an eye on the brake fluid level in the master cylinder reservoir so it doesn’t drop too low and pull air into the brake line. Continue the process until you see clean brake fluid drain into the container. Tighten the bleed screw, top off the reservoir if necessary and install the cover.
Now raise the rear of the sled on a jackstand and test the brake system for proper operation before you hit the trail again. Inspect the snowmobile and area where you worked for brake fluid and promptly clean it up. Brake fluid is very corrosive.