This is an absolutely fine plan, and I don't think this will jeopardize your candidacy in any way. However I would recommend that you mention in the additional information section that you are keeping your intentions confidential which is the reason that your supervisor is not one of your recommenders.
Make sure that everybody is telling an anecdote firsthand that shows they have observed you demonstrating leadership teamwork problem-solving and initiative.
The additional recommendation is truly additional you don't need to have three people, and it's actually a little better for you to wait until the middle of the process may be around interview time for you to bring this one forth.
It's ideal for you to have an alum of the school but I think your idea is also good. However it's also a test in judgement!
Meaning, it should only be someone who is bringing a different angle to your application supplementary to the rest of what you already have.
Don't move forward with a third recommender just to fill the space that's not going to work to your advantage.
https://mbaprepcoach.com _________________