Joel,
you are going to want to free float that barrel, then bedd the action.
that will relieve any pressure on the barrel, what bedding does when done properly is perfectly set and center the action allowing for a stress free foundation, so when removing the action for cleaning and return to the stock after your action is now custom fitted allowing for repeatability.
So yes it is highly recommended and will only improve the preformance of the rifle.
The ruger 77 is a tad more difficult to attempt, if this your first bedding job I would leave it up to your local smith, but if you wish to attempt it your self go slowly and use plenty of release agent.
good luck!
RON KONTOWSKY
DEANS GUNSHOP II