My brother's combat mg mk2 has a scope though.
Well let's decide which weapon we're talking about first...
Combat MG Mk.2
It shouldn't really matter which way, but you're saying stuff that contradicts the information you've already provided - "His character holds it closer to the hip like this. <video link>" shows the Combat MG without a scope, hence that stance.
If he's in stealth mode, as pictured above it'll be a mid-chest aim, but you need to remove the scope to hip-fire any of the three above.
The only other thing is if he's in first person, but again it won't be hip fired. This is only how it looks when in first person but not aiming. If he starts aiming, the weapon's brought into the shoulder as usual.