Rem 700 LSS in left hand. Available in 300RUM, 340RUM and many other cartridges. They are available for less than $700. This would leave you plenty of money for customizing if you felt that it was desirable.
For a moose and grizzly rifle I would think that you would be looking at 338 mags and up. Lesser cartridges can work under ideal circumstances but I wouldn't "choose" them for these game animals.
Check out this link for a look at the available left hand rifles. Left Hand Rifles