You are trying to cover too much with one rifle, period.
You want a gopher gun? Start at MOA(minute of angle) out of the box or less in a .204Ruger. .223Rem or .22-250Rem etc etc, and then for Moose Season, you can go with any appropriate bullet in a 7mm-08Rem, .270Win, .280Rem, .308Win, .30-06Spfd, 7mmRemMag or .300WM/WSM.
To meet the requirements of a Varmint rifle in regards to Minute of Angle or less accuracy, and won't strike more than 3" over your line or sight or 3" low, out to 300 yards is a tall requirement. To meet the requirements of a suitable Moose rifle, it needs to develop the energies required to drive heavy for caliber bullets deep and expand well in a moose.
You see you really need two rifles here or a
Jack of all trades/Master of none compromise, which I know you seekth, but will do neither of the two jobs perfectly. So here are four options that will do both, but with limitations to the basic criteria of a true Varmint and a true Moose rifle: You choose the platform and scope based on your budget, there have been great combos listed already..............
260Remington. Cost per round acceptable in varmint use, minimum acceptable trajectory to make a baseline pure varmint round combo cartridge, and bare requirements for energies and bullet weights for a baseline Moose rifle. Out of box accuracy for small targets close to Varmint requirements
7mm-08Remington. Costs per round is acceptable, especially for Moose, and for light to moderate Varmint use. Recoil is acceptable, and barrel life is good in the varmint role. But you will have to deal with the steeper trajectories during long range shooting on a very small animal and range error estimations. Windage however, will will be more acceptable. Good for Moose out to 200 yards.
.257 WeatherbyMagnum

. Cost per round is debilitating, even to a handloader. Will shoot as flat as a pure varmint rifle with lighter bullets. Windage drift similar to a pure varmint rifle. Will develop excessive muzzle blast, recoil, and barrel life will be very short in warm weather extended firing on gophers. Develops good energy and penetrating power with heaviest bullets in a moose. Good for Moose 200yds+
.308Winchester. Cost per round excellent with humpteen bullet weight and type choices for field and practise use lowest cost per round in factory ammo. Recoil acceptable to most, but starting to get there in a varmint rifle. Barrel life good, but again the trajectory is going to pose a problem on animals standing only 1 foot tall or less at distances out to 300 yds plus. Wind drift will be good, similar to the 7mm-08, but starting to widen on small target now. Good for Moose out to 200 yards.
So if I had to pick one caliber for Gopher and Moose in the purest sense trying to meet both's criteria well and firing
one shot at each target per year, I'd take the .257WeatherbyMagnum.
But of course you are not going to buy the Weatherby for gophers, it's perfect niche is plains shooting on medium size game, so hence your dilemma of trying to do too much with too little options.
Buy two rifles, and save your self the heartache of trying to do so different a role with a one caliber compromise, or pick from the 260/7mm-08/.308 based on each virtue listed. You will enjoy both alot better.
Compromises in firearms are great in theory, and they work well in military or tactical applications where the threat or the environment can change by the second, but a game hunter usually chooses his targets and engagement distances on his terms, so the job specific rifle is always a much better deal for him.
Man I must be bored.................
