That's one impressive claw!
Leaving the claw outside will allow the flies and other bugs to eat the stinky meat. It may take a couple months before all of the meat is gone (depending on temperature, etc). Don't leave it on the ground or bury it because the shell will start to demineralize where it touches the ground, and you'll have a hole or a rotten spot on your trophy.
Don't leave it in the open where a raccoon or possum can tear it up. Put it in a minnow trap (a cylindrical metal cage) or something similar to protect it from animals while allowing the bugs to have at it. Put it far enough from the house and your neighbors that the stink won't cause problems.
After the bugs have done their job, pour hydrogen peroxide (3% solution you buy at the grocery store) into the claw. The H2O2 will clean it up nicely and ensure that all of the stink is out of it. However, H2O2 will bleach all of the color out of the claw, so don't leave it in for more than an hour or so. After the H2O2, rinse the claw really well with tap water and allow it to dry overnight. If it still stinks, try the H2O2 again (sometimes it takes a couple treatments). Once clean, if you soak it in water for a while, you may be able to reposition the claw, but I wouldn't count on it.
When it's thoroughly dry, spray it with clear-coat polyurethane (I use matte finish so it isn't glossy), inside and out (be careful spraying it inside as it will blow back at you). Follow the directions, and give it a few coats. Once it's dry, it should last a long, long time.
You're best to leave it empty, I think trying to fill it with expanding foam will likely destroy the claw.
There's a great web forum for taxidermy (taxidermy.net). They discuss all kinds of gross stuff, but they have great info.
I hope my reply isn't too late. Good luck!