SwampCollie

A lot of this answer depends on whether you are handloading or not. I'm guessing you are if you are considering an AI cartridge (and yes, .243 Ackley rifles MUST fire standard .243 Win in order to fireform brass.... so you'll be shooting .243 win out of it initially.... then reloading the new Ackley 'Improved' fireformed brass in order to achieve greater velocities).

The hard part about what you are looking for, is anything lightweight is going to have a very lightweight barrel. Having a barrel that thin, and getting sub-MOA accuracy out of it reliably is no mean feat. Granted, you are after the first well placed shot, which is as it should be, but follow up shots start walking VERY fast with light barrels... which is why its hard to make a light weight varmint gun for volume shooting. It just all depends on what kinda varmint hunting you do. Lets stick with speed goats for now just to make it easy:

The 6.5/284 is largely a benchrest cartridge.... and its not a .243/6mm.... a 6.5 is a .264 which is a fair piece larger when it comes to small caliber centerfires. The 6mm/284 is one hot round, and in a light barrel... its going to be tough to control in my opinion. I'm not a gun builder, just a tinkerer, so you may well have an expert on the subject tell you otherwise. I am a fan of the plain old 6mm Rem... sadly, not very many rifle makers are. Not unless you are buying just an action and having a custom barrel made for it. Which you certainly can do in any of the calibers listed.

Since lightweight was the first adjective you gave us.... I'd recommend a Kimber Montana in .243 Win. It will not be terribly expensive to have it reamed to .243 Ackley Improved. I own a 30/30 AI, and it shoots the standard 30/30 win rounds extremely well, though the point of impact difference even at close range is 'huge'. It would, however, in a pinch give you the option of shooting inexpensive .243 ammo for practice or varmints, while shooting Ackley handloads with premium bullets for deer/goats whatever.