Sorry, and you might stop reading after this first sentence based on your belief that those without allergic kids don't "get it," but my kid doesn't have food allergies; however, I am on staff at a camp for a week each summer.
Any camp or course or class has its rules, they are in writing for all those who apply to see them, and there for you to consider as you enroll your child -- or as you decide not to enroll. If you want to talk to the camp director about the reason for the policy, do -- they are open to questions and you have a right to ask. Ask if their policy is due to insurance, or staff training, or some past experience, and if they will make an exception. But be aware that they can say no; they are a private entity and make their own rules. Public schools are required to make some accommodations--but a privately run camp isn't necessarily required to make the kind you're wanting.
Making a change that accommodates just your child could do the following: Create liabilities. Void their insurance policy. Require additional staff training that they do not have time or staff to do this late in the game.
And if and when other parents find out the camp's protocols for one child are different from the rules sent to all parents, the camp will end up spending a lot of time defending that one exception, and explaining why another parent's child didn't get the same treatment.
Of course it would make sense for them to do the Benadryl first. Talk to the right person, high up enough, and you may be told, "Sure, we can do that, we've done it before, glad you asked!" when you inquire about it, and you'll be happy and they'll be fine with it. But for your future reference, you need to know that you'll run into this again.
You seem to be baffled that a camp run by a "top 20 university" for gifted kids would not accommodate your specific needs. But it's still governed by liabilities, insurance, staff and training requirements and rules -- which apply equally whether a camp is for gifted kids or not.