Not sure if you like hunter-style stock better or thumb hole...
The Axiom is very light but quite flimsy and the pistol grip is like a kid's toy imo...
Have you ruled out a folding stock...
I can't account for which brand is "best" but I did look carefully at both Whistle Pig and Tactical Solutions aluminum barrels...bought the Tactical Solutions because it was off the shelf...literally...at sportsman's warehouse...and cost slightly less...very slightly...but if having the most accurate possible or a compensated aluminum barrel and willing to go to a little more trouble to get it I would have gone the whistle pig route...
The barrel weight is night and day...my factory 18" barrel was 28.5 oz and the TacSol barrel was 14.2 oz...the barrel length and weight are far more important imo than the stock weight...balance goes a long ways with regard to the "felt" weight...
Mine is currently 4 lbs 13 oz with a loaded 10 rd mag and iron sights... 5 lbs 6 oz loaded with a Leupold 2-7x28 scope... mounted on Weaver TO-9 rail with Leupold Rifleman low rings...
Its 32" long overall and I am sure I could knock a squirrel out of a tree at 100' one handed...that is because it has great balance...a short length of pull...and really large and easy to hold thumbhole pistol grip...
For $70 from midway you can get a power customs adjustable sear and target hammer and have a trigger pull in the 3.5 lb range...and adjust out all the slop...but it won't completely eliminate creep...so I don't know how "crisp" you have to have or how much you are willing to spend...
One more thing to consider... Tactical Solutions makes an even lighter barrel than mine... it's the SBX...not sure exactly how much lighter...you will NOT like the price...!!!
Also...I want to keep my stock NRA SB match legal...but if I didn't...most but not all of the fore end would get cut off...that would take another 1/2 lb off and vastly improve the balance at the same time...
With a good thumbhole stock cut down to your liking... a 2-7x28 or smaller scope...and an SBX barrel it would be very compact...very light...absolutely under 5lbs loaded...I would say less than 4.5 lbs is very doable...easily used from the shoulder with one hand...no problem...
Take your time and look around there is no shortage of options...