I don't see any bridges being built here.
As interesting as your idea is, and believe me I have thought on this, having skirmishers coming in and out of a Milsim game could be disruptive to persons already involved in an extended scenario.
An example. If a designated Milsim game is into it's 14th hour and opposing teams, who have been briefed inside and out on their specific taskings and know them like the back of their hand, have taken the utmost care in achieving their objectives and are operating in a sensitive environment, have to halt the op so a player coming in can get up to speed on all the op specifics? Without rehersals? How will he find his team that's already balls deep into the op? What stops this new player from jeapordizing the whole mission because he's decided to "camp out" in an area you've already negotiated and passed undetected? That may come off as too much bullshit, but those games exist, and many thrive on them.
Flipping the coin here, I'm sure you've seen first hand how players sigh and moan when a milsim guy starts on them about comms and formations and fire discipline. You wanna say, "Dude, join the army then." Milsim guys will take the freedom out of a loose skirmish by trying to organize the shit out of it, and players will not feel like they are in control of their own actions any longer.
I admit that I adore Milsim, and the people it attracts, and I'm not one for skirmishes, but I've been known to skirmish from time to time. When I do, I respect the rules and understand that people aren't going to follow me because I've been made a team leader, or whatever. It's a bit of a free for all, but still team vs team. In the end, I have a hard time making the transition. I am led to believe that it is the same with skirmish players who enter Milsim Ops.
If that's the case, why not please both sides with their own sections? How do we know it won't work unless we give it a try?