I apologize in advance for the length of this post but rest assured that this version IS the condensed version...lol !!!
I don't think that removing the first two chicks right after they hatch is going to be your biggest concern. It's going to be the clutch of unhatched eggs and the length of time that remains before they're scheduled to hatch. Like others have said, once your hen notices that some of her eggs have now become chicks, she more than likely won't continue to sit on the rest of the eggs for more than a few days, maybe a week at best. Call it hormones or Mother Nature or sublime chicken intelligence, but whatever you call it it's what makes hens get up and off, basically abandon, the eggs they're sitting on after a certain period of time. Even the broodiest of hens, when faced with any unhatched eggs after a certain period of time, will abandon them and start over with a fresh lay of eggs. They just do. If you were only talking a couple of days between the first and second hatches, you might be successful. You might be. But with an entire 2 weeks between them I just don't think it's going to work no matter how you try to juggle it. This is what I would do if I encountered your situation but please don't take it as fact or as a fool proof way of getting all the eggs to hatch, it's just what I would do.
If you don't have other broody hens and if you don't know anyone with a broody hen that you could stick the newest clutch of eggs under then I would buy or borrow an incubator and put the two original eggs that your hen layed in the incubator until they hatched and leave the rest of the eggs (or the second clutch) under your hen until they hatched. I wouldn't even try to stick the first two chicks back in with your hen until all the others have hatched and even then I would do so very cautiously in order to make sure that your hen will accept them hanging around what she has come to know as her babies and not try to peck at them or chase them off. Others on here may disagree but here is my reasoning on why I would choose to do it this way.
1. the fewer eggs I have to mechanically incubate, the better off I know I'll be and most likely, so will the eggs (chicks).
2. the fewer chicks I have in the brooder, the better off I know...(see rest of explaination above in #1).
3. I like to gamble but I'm not a big risk taker and so I usually play the odds and the odds in this situation say that it's better to risk losing the first two eggs (or chicks) than it is to be almost guaranteed of losing all of the second batch of eggs (or chicks).
But like I said in the beginning, this is just my thought on how I would handle your situation should I ever happen to encounter it. I wish you the best of luck and please let us know what you decide to do and how it turns out as I'm sure others on here would be interested in knowing as well.
P.S. as far as putting the second group of eggs in the incubator should your hen abandon them...You could try to do that but you would have to have your incubator up and running at the proper temperature and humidity level in anticipation of that happening well before it actually happened. Not to mention the fact that once you put the eggs in the incubator, the minor temp and humidity adjustments you'll have to make might just be too many variables or fluctuations for the eggs and they may no longer be viable because of this. I don't know about anyone else, but it takes me several days, at least, to fine tune my incubator before it's ready to accept any eggs. (Any eggs that I actually want to hatch that is....lol) If you don't have it up and running, in the time that it would take to get it that way, you will have lost the chance of them hatching do to the eggs being cold for too long. Good Luck !!!
