I would vote the Savage route. If it's truly budget oriented and you're OK with a factory sporter barrel, look at the 111/116 Trophy Hunter XP. 111 is black, 116 stainless. Right around 500 and you get DBM, old Nikon Prostaff 3-9 BDC scope, and decent factory trigger. Most report at least 1 MOA with the factory Savage.
Worst case, try it out. If your not happy with results, sell the barrel and rebarrel it. That puts you $400 into it after the barrel sale. You can EASILY get the action trued and a quality barrel for $600. You'll likely have enough left for a replacement stock. Boyds started making the DBM stock for Savages if you like laminate. They offer Thumbhole, monte carlo, standard, and their tactical models.
The scope is def not top quality, but it's serviceable. If you have one, you can always sell that too for a few bucks.
I realize this is outside the box and NOT necessarily the best option. Just throwing it out there in case you like to tinker. Since you're new, you may not have thought about this route.
If you're not set on a .300 WM, I'd look at other chamberings. 500-700 is doable pretty easily with your current .243 as long as it shoots. 300 WM is a pretty large jump recoil wise as well
6.5 Creedmoor will get you pretty easily to 1000 with very little recoil. Hornady factory match ammo is also very respectable and reasonably priced - usually around $25 +/- a box.
Anyways, just throwing out some ideas since you said you're new to LR shooting. Not trying to talk you out of the 300, just put out some ideas in case you hadn't thought of them.
Chris
P.S. I mainly point those things out because I was set on a 300 WM for my first LR rig. Best decision I made was to go a different direction. 300 WM is just recoil overkill unless you want to push 1500 yards. 7mm mag runs RIGHT next to the 300 much more enjoyably.