I did basically this same trip with a stop in Paris several years ago. We did fly into London and out of Zurich. Yes, the fees are a little higher in and out of London, but the fares are often cheaper so it averages out
(I often fly personnally into London as from where I live it's often the cheapest option)
We did wind up using a TA who got us a good group rate on Delta. If any of your parents have a contact with a TA I would see if they can help you.
The only person who used FF miles turned out to be me! So that wasnt' an issue. IF they parents want to use miles (which we have had on other trips) there are some issues. Once I say that it's basically the "girls job" to get to the first hotel on time they often give up on that idea! You really can't be running back to the airport every day for the stragglers!
I would definitely check Icelandic air if it's an option for you. They seem to have the best deals.
A couple of other notes.... I assume you are going to stay at World Guide Centers. In London it's a great deal compared to the hotels and in Switzerland it's an amazing experience. Depending on how long you are gone you can pay them to do laundry at Our Chalet. Best money I spent LOL (Another troop that was there with us had wet blue jeans hanging around for days because they couldn't afford the laundry!) ANd that brings up my 2nd point. "build a pad into that budget" It's better to come home with funds then to be paniced over things!
Do not include the middle flight in your booking. Trust me it's probably going to mess you up
Just do as you plan and buy the London to Zurich seperately (you can also do Geneva. WE wound up doing that and it worked well)
Luggage! Somehow you need to make sure they pack lightly. There's a lot of hauling around and the best discounts often come on airlines who charge a lot for the steamer trunk. I failed on that but the girl who got "busted" on the flight from Paris to Geneva had to pay the extra cost out of her spending money.
ALl of my girls said they wished they had packed lighter and were jealous of the two leaders that did get everything (including uniforms, boots etc. ) into our 22" bags.