We called it Yankee Swap, and I've never played it with a limit on the number of swaps. The thing to do ahead of time is decide if it's a "white elephant" game where people have to bring something from their closet, or if it's for new things and there's a specified dollar amount ($10-$15 or $15-$20 is common). That way there is never a "cheap" gift in the mix.
The problem with teens is that they don't just "covet" but they also are so sensitive about rejection and not being "cool" - so a gift that gets chosen frequently kind of rewards the person with such great taste, and the one that no one wants to keep usually only gets traded away once (it might not get "chosen" again but it can be re-traded). It depends on how good the group is about being supportive. I've been to swaps where the outrageous laughter over a funny gift is part of the entertainment - one year someone gave a tube of "Crack Cream" and there was plenty of adult laughter about that. I forget what the rest of the package was to make it $20 but no one else remembers either. Besides, a lot of times people want a gift that is perfect for their sister or mother, vs. for themselves.
Limiting the trades means that the people with higher numbers don't have as many choices, but then again, the person with #2 or #3 didn't have much chance at all. The person with #1 has the most chances because she gets to swap at the end (as the opener of the first gift, she just sat with it and had no options about trading at all. So the point of it all has to be about the fun of choosing and the mystery of opening things and the laughter, rather than the "what can I get for myself" philosophy - all of which fits in with your scouting philosophy.
You could also ask people to purchase from thrift shops that support a charity or from a small business - that puts a "community service" spin on it. People might wind up making a $15 "package" out of 3 or 4 purchases of $4 each, which means every "set" will have something of value and interest to every girl.
Another feature you can add, instead of the numbers, is "The Wright Family" - google it. It's a story about a fictitious family, and it's loaded with the words Wright or right, and left. Everyone sits in a circle with a wrapped gift on their lap, and someone reads the story (could be the scout leader or anyone with a voice that projects). Every time the word right (Wright) or left is mentioned, the group passes their gift to the right or the left. So the gifts wind up rotating a lot and no one ends up with their own gift. (If they do, at the end, the leader just says to pass the gift to the right.) Then they open what they have. That might meet your needs of everyone being grateful and no one coveting.