The British manual that you have posted:
A- What’s the date on it?
B- It clearly says it was designed to be 327 lb. “327 lb (Designed)”.
Not to mention that it doesn’t even specify any variant …
Spoiler
I.e. This is some generic weight that they were initially aiming for in the design process of the missile.
This is not the actual production weight that was actually realized for any particular variant.
If you look at US aircraft manuals, they all have AIM-120A listed at 348 lb:
https://community.gaijin.net/p/warthunder/i/c3Zavuod4L8o?comment=2Sy4bxZKtZnMAxSUOheBfuHR
Note that I’m not using these as source as I don’t know if they are declassified or not.
But I have provided two public primary sources and one secondary source:
1- “US Defense Logistics Agency Public Logistics Data”: 342 lb
(I don’t see how you can get more primary than this)
2- “ADA357045 Distributed Simulation Testing for Weapons System Performance of the FA-18 and AIM-120 AMRAAM 1998 LCDR Tom Watson Naval Weapons Test Squadron”: 345 lb
(People in charge of testing and simulating the weapon’s performance in 1998 got the weight wrong! right?!)
3- “Encyclopedia of Modern U.S. Military Weapons 1995 COLONEL TIMOTHY M. LAUR AND STEVEN L. LLANSO Edited by Walter J. Boyne”: 345 lb
These are primary sources from the US (which actually makes the missile) whereas the undated UK document that doesn’t even specify the variant is clearly not as it explicitly mentions that the 327 lb is what the missile was initially intended to be in the design process, and not the actual weight of missile that was actualized in reality.
It’s also possible that the 327 was the propaganda/fake figure which was given to magazines etc, as magazines in 1985 and before (before the missile actually went into production) all use that figure: https://community.gaijin.net/p/warthunder/i/c3Zavuod4L8o?comment=xaNLwmEChsaYsMAXSVfHwob7
Regarding this report: https://community.gaijin.net/issues/p/warthunder/i/c3Zavuod4L8o