You want to get started for under $1000, which is fine.
Just to keep your eyes open on this, how much do you expect to shoot, i.e. how much do you plan to spend on ammo? 250 rounds a year? 1000 rounds a year?
I'm not suggesting that you spend more money on your rifle, but you should be aware that ammo consumption tends to be more expensive than the cost of the rifle. Even if you handload, 1,000 rounds per year of .308 will cost about $600. A thousand rounds a year of factory match ammo would simply not be affordable - I'm likely out of date here, but it could be $2,000 to $2,500.
If it is at all possible to set aside your requirement to use the rifle for moose hunting (zero to three shots fired in a year), that would greatly improve your options for target shooting (hundreds of shots fired in a year). If possible, I would suggest you choose the best rifle for your target shooting requirements, and then either buy another rifle for moose hunting, or borrow one, etc.
I have shot a .308 target rifle for 16 years. It took me three years to (mostly) get over a flinch, I have struggled to keep it away since then, in fact I am still not over it and still have to constantly fight to keep it away. Last year I was fortunate enough to win the most prestigious match in Canadian fullbore iron sights target rifle shooting. And at 900m, on the third-last shot in that match, I flinched, and lost a point, and figured then and there that I had lost the match. So please believe me when I say that I wholeheartedly agree with Ian's advice that a .308's recoil slows down your progress as a shooter. If I were God and I were inventing the "TR" game from scratch today, I would likely choose 6BR as the cartridge to be used (due to military heritage, we use the military cartridges - .223 or .308).
Your best bet would be a .223 or a 6BR, with the 6BR probably being better and more versatile (especially if you want to go the barrel-swap route to get a moose-capable rifle). Both are somewhat cheaper to shoot than the .308 (the bras cost is similar, they burn 55-65% as much powder, their bullets weigh 50%-65% as much as the .308's and to a certain extent the bullet prices reflect some of this). More importantly, they both have meaningfully lower recoil than a .308, are pretty straightforward to load to deadly accuracy (as is the .308), and with the right choice of match bullets (80 and 90 grain bullets for .223, 105-ish grain bullets for the 6BR) are able to match or exceed the .308's wind drift characteristics (which is a principal factor in long range shooting performance).
I would suggest that you avoid a muzzle break. It is unnecessary expense, it is not allowed in some competitions (they are brutally loud), and especially with a .223 or 6BR it isn't necessary.
As others have mentioned, target grade iron sights are fearfully expensive (on the order of a decent scope). And hunting style iron sights are not really suited to being used at a variety of ranges where sight adjustments will be needed. I would suggest a "good cheap scope", such as are available. This will actually be a more difficult thing to get right and make the right budget compromises on, than the choice of rifle.
Nothing wrong with a detachable magazine, it's a nice convenient feature, but absolutely no need for it either. For a target rifle, single shot actions are preferred (they are simpler, and also stiffer because they don't have a hole cut in their bottom). Besides, once you learn how to shoot with a target rifle, really learn how to shoo, you won't suffer any crisis of confidence going hunting with a single shot rifle. You'll *know* and understand that the shooting part of the hunt is something that you've already mastered.