.308 is the most common. Generally we don't see a lot of people with their own .308 get a job with a police force and use their own rifle as a service rifle.
Depending on the competition that you are planning on one day entering, that would greatly play on the caliber that you are going to get.
1,000 Yard Benchrest - you don't want a .308 or a 7mm Mag
Across the course gun? Possibly a 7mm mag, but that is a lot of weight and/or punishment to be lugging from station to station.
The FBI HRT Team is currently fielding 7mm Rem Mags on a Remington Action a 26" heavy barrel. But, they are usually green lit at longer distances than your normal police sharpshooter.
Traditionally, the average of S/S involved police shootings are around 71 yards - though the last stats I saw were from 2006, so that may have crept out a bit, but I doubt it's past 90 yards.
At that distance the 7mm Rem hasn't even calmed down yet and will likely take the head clean off a target. I would go with the .308 for these ranges. It's common, it's easily defendable ( in court ) and it's what everyone else is using. Not to mention you are already familiar with the ballistics of the round.
In a factory, barreled rifle action, with a warranty and an accuracy guarantee, I would be hard pressed to put you in anything that can touch the Weatherby Vanguard / HOWA listed above. Out of the box tuned performance, plus you get accuracy with factory ammo and don't need to hand load right away. That is a good bang for your buck option.
A custom build is going to run you $1,000 ( or more depending on the 'Smith )plus parts - anything like fluting the barrel, special bedding or custom stuff ( muzzle break, weight management on the stock ) is going to run more. For $1K plan on getting your choice of products put together, the action blue printed , a trigger job to your specs, your optics mounted in place and someone to give you some basic advice.
In the projects section you will see my Winchester Sniper Rig in .308. I had that Krieger Barrel purchased ( $ 300 or so ) and then had the chamber cut ( $350 ) specifically for the specs on Federal Gold Medal Match ammo. That way I don't have to reload, my chamber is cut specifically for the specs of the off the shelf ammo. It shoots sub MOA all day long and will shoot 1/4 MOA when I do my part. I also have a lot of extra toys on there that add up pretty quick.
I think the best bet would be set a price that you want to spend on the rifle - do not include the optics, because that is another headache, but how much do you REALLY want to spend on the rifle? From there, I am sure we can get you in the ballpark with something you will like....
JD