The only difference platform makes in this instance is cost of components (and avaibility). The 338 - 7 BR is a great round BUT would be more suited for the AR-10/DPMS LR rifles. These are more costly and components less easily available (typically). The Spectre fits in the AR-15 which means parts can be a little easier to come by. The Spectre also operates at lower pressure and less powder than the 338-7BR which will aide in suppression.
Shooting beyond 500 yards can be done, at a military T&E a .50 cal Whisper engaged the commander's hatch of a MBT at 900+ yards with amazing accuracy. TOF is to the point where you can put the rifle down, pick up spotting optics, drink coffee, pee, read a magazine and then witness impact
Note the similarity in trajectories of the 300-221, 338 and the 510. Something you then need to consider is energy on target, which at subsonic velocities becomes a matter of bullet mass. The winner in this case is the .510 (as 1000-gr projectiles are readily available as are Raufuss ... right Grinch?
) Problem with the .510 in a truly "quiet" role is that the HUGE projectile actually becomes far more audible in flight (ever here Might Mo's trash cans fly over?
or even 105 for that matter) The smaller 330 and 338 will be better in that regard.
Something to consider: The larger Whispers use the BR case, and from the pictures you can see how much bullet protrudes. Were I to do something similar on the AR-10 platform, I might consider the IMHSA version of the Savage case as a starting point so less bullet is "hanging out". The 7mm variant itself is a consideration, with 250 gr RBBT Spitzers to be found.
Either way, Randall and I can set you up and not charge you for the privilege of using our stuff like someones does. Randall does great work and he and I need to continue our chat on fun stuff. Darn time difference ...