I have a couple double rifles, and like them a lot. You have been given some good advice here, but be cautious about some of what you receive because double rifles are the subjects of passionate debate. There are two basic "schools" of thought - those who think the British are the beginning and end of the double rifle culture and history, and the European "others" who are so gauche that they actually promote the use of modern materials and methods, and over/under configurations, and perhaps go so far that they hunt with a scope attached their double rifle.
I like both. This is what I did:
I can't abide the asking price for a "proper" British double but admire their artistry, style, and history. But $20,000 and up is too much for me. So I have a solid and serviceable German Merkel 140AE that shoots a grand old British cartridge, .450-400 3" Nitro Express. ( .400 Jefferey) My eyes don't agree with iron sights any more so I have it fitted with excellent Recknagel QD scope mounts and a nice 1-4x Kaps scope. It shoots accurately, handles well, and efficiently killed the Cape Buffalo and bull Elk that I have hunted with it. I have under $10,000 invested.
I also admire the European tradition of hunting driven game with a quick handling medium bore double rifle. Having hunted in Germany a few times broadened my horizons a bit. It's not only the Brits who know how to make good doubles, no matter what so many Americans or Aussies say. The Germans, Austrians, Italians, etc. are not so constrained. So they make some very lively, accurate, and relatively affordable O/U double rifles specifically for hunting Battue, Druckjagd, or whatever you call European game driven to the gun.
Mine is a Beretta Silver Sable O/U in 9.3x74R. Based on their proven technology and built on a 20 ga. frame, it is much more lively and responsive than my SxS. It shoots quickly and accurately with the fibre optic open sights at close range, and very accurately with the S&B 1.5-8x42 illuminated scope at any reasonable range and in any light. The fallacy of an O/U being too slow to load is just that. A story made up by anglophile sxs double lovers. I've used mine for a couple of local bears so far. That rifle is also under $10,000 all in. Beretta doesn't make the silver sable any more, but several European makers do make something similar.
My advice is to stay away from single triggers, double triggers are more reliable, more traditional, and always work. They do take a little getting used to. If you like the super modern approach taken by Blaser and a few others, there is nothing wrong with that. Function trumps style, as long as you like the style. Be aware that double rifles in side x side configuration are challenging to fit with a scope. A scope can be used, but the barrels may not regulate well if the rifle was not fitted with a scope at the factory. One of the great virtues of modern European medium bore doubles is user barrel regulation, via a screw/wedge arrangement near the muzzle. That option is only available on guns of moderate bore size, usually 9.3 and below. It allows use of scope or no, or different ammunition, with accurate convergence of both barrels at point of aim. If you can't buy the exact ammunition that a double with fixed barrels was regulated with, be prepared for a hefty bill to re-regulate or a long process of load development. And rimmed cartridges are the way to go. A good double can be made for a rimless cartridge, but it's a mismatch of technology in my opinion. Of course I'm biased!
Your goal of finding a "good" medium bore double for $4000 or so is admirable, but not really realistic. Do your research, save some more, and get what you really want. You will be happy!