I want to be very explicit with my answer. The others before mine are reasonable at getting at the generics, but there are a few things you're overlooking.
My current role is somewhat critical. Things won't grind to a halt without me, but it will be a pretty big hit, and current projects won't complete on time. It would also probably be a pretty big morale hit to the team, but an extended notice period might not help there.
This is a double-edged sword. Fundamentally, you have no real guarantee that you are super critical to a project. A reasonable team would have the ability to fill in for you if you were out on vacation, and your departure could be translated to "indefinite vacation". Other things like morale aren't really in your scope of control, since the problem exists at a level above your presence anyway.
The complication, as I see it, is that pretty much all advice relating to leaving a job is clear that you shouldn't say anything until you have a signed offer in hand (which I understand and agree with), but at that point, I would already have agreed to a start date with my new employer, which means my current employer has no incentive to engage in the negotiation.
This is incorrect. The best practice is yes, don't bother leaving or saying anything until you have an offer in hand, but this isn't where negotiation ends - it's the only logical point where negotiation can even begin! Your current company would now understand that you were serious about leaving, and that you have an offer for X more than what they're paying you, and if they believed that they could invest in you, then they'd match or exceed the offer you got.
At that point, you could go back to your tentative new company and say that they provided you with a counter offer, and that you're looking to stay. This gives them a chance to counter that.
But even then, you're still on the winning side - either your current employer counters your new offer and offers you better wages, or you start a new job with better wages. Only thing that could cause that to fall apart is if something really nasty came back in either a background check or financial collapse at your new employer. Both instances are rare, but rare is not the same as "cannot occur", so bear that in mind.
How do I bring this up prior to having an offer in hand (and preferably, before I have interviews lined up)?
You don't have anything to fall back on to, so you don't have any room to speak up or to make a bold declaration. You risk your employment on having no agreement in place beforehand. Until you get an offer or until your company decides to initiate a conversation after seeing why you're taking a lot of vacation time at odd times, there is absolutely no benefit whatsoever to even talking about it.
Go out there, do the interviews, and secure an offer. Only then do we start talking about departure periods.
Oh, by the way...
The two-week period is meant for the company in that they have two weeks' notice to start the HR engine to begin hiring for your replacement. If the team wants more of your time than that, then you should negotiate in clear writing how long it will be and how much you'll be compensated for it. If neither party can come to an agreement, then it makes sense for the original agreement to stand, and you'd leave that company at the end of the two week period.