You guys are missing the point.
I want to condense two rifles into one in a compact lightweight package. I just want to know if anybody’s handled one.
I'm going to suggest that .338LM and light are not exactly going to make you a happy shooter if it's for range work / target work, or anything other than humping it to go hunt something where you might take 2 shots or such. The same is true for other rounds in that class such as the .338NM or .300NM
My suggestion is not to try to do it all with one rifle if you have an actual need for something light sometimes and a need sometimes to shoot stuff in the super magnum class.
An AXMC would be a very nice gun, that runs everything from your 6mm to your .338LM class cartridges.
You can get barrels for it from lots of sources and it is well proven.
Don't discount the importance of being able to easily source barrels, that's huge!
You'll come in around the 20 to 23 pound range if you are fully kitted up with a heavy & long .338LM barrel, depending on your scope and bipod.
If you have a light, short barrel in a smaller chambering, with a light scope and light bipod, you can get under the 20 pound range.
Then figure out exactly what you want a "light" rifle for and get a rifle specifically built for that, most likely in a dedicated short action platform.
Things like the SIG Cross and such are purpose built for humping a compact rifle long distances for hunting and such. If you want your "light" rifle mostly for doing PRS style games, then just get one specifically built for the gaming you want to do.
Now that being said...
IF you are interested in pushing the limits of ballistics and distance, you should seriously look at the upcoming Hoplite rifles.
The Aspis which is nearing production is going to be first out in the .338LM / .338NM / .300NM boltface and then there will be options for going up to .375 Swiss P (same bolt) and then going down to short action cartridges with a simple (tools required) barrel change, bolt swap and different magazine.
The advantage of the Hoplite rifles is they are built to tolerate much higher pressures (in the 80k to 90k range) for regular use, than other rifles, so with the correct brass / cases, you can push things like solids much harder than you could with somebody else's rifle.
The weight and price is going to be similar to the AXMC