A useful sniper rifle is going to weigh a lot more than a useful AEG since the sniper rifle needs to be upgraded to the nines to be effective. All AEG rifles are light in stock-form, honestly, if you can't pick one up and haul it around with you all day then this isn't the sport for you.
Airsoft is a very physically demanding sport, if you're going with a sniper rifle so that you can sit in one place most of the game and "snipe" people you're going to be in for a very bad experience. In fact, airsoft "snipers" need to move around even more than regular players to be effective. On top of that they don't get nearly as much action as "frontline" players because all the moving that airsoft snipers do is to stay out of sight and far away from the enemy players (see: running around the edge of the field).
Everything you've said shows that the information you've researched is either wrong, or that you disregarded a lot of the really important stuff.
If you want a good sidearm for "sniping" and you have big hands, the ONLY option for you is a TM MK23 NBB. It's a god-like pistol for snipers. Anyone who has owned one of these and reads this comment will know what I'm talking about.