The danger here is that your additional terms might be ineffective. The problem is, if you distribute it to an individual or a non-profit organization under the terms of the AGPL, then that individual or non-profit organization has the right to redistribute the software to anyone, including for-profit organizations, under the terms of the AGPL. It says so in the very first sentence after the copyright notice!
So if you do not want to allow this, then you must use something other than the AGPL. (Or maybe you could do some funky patching of the AGPL's terms, but you'd have to be careful to only do so by reference, since its text is copyrighted by the FSF, who only permit distribution of verbatim copies.)
Also, you should probably avoid calling it "fully free & open source" as you did in your original comment, since you intend for it to be neither Free nor Open Source in the sense ordinarily meant by FOSS.
In all dual-licensed FOSS projects which require licensees to comply with the terms of both licenses, both of them include an unlimited right to redistribution. Also, roughly no projects do that with the AGPL, since the AGPL includes the right to cast off any additional terms when redistributing.
But your intent, if you want to avoid for-profit organizations using your software, is to impose a restriction on users not to redistribute the software in certain ways. It is your right to add such a restriction to your software, but it is no longer FOSS, since that term necessarily implies an unlimited right to redistribution (within the law).
Your first claim is provably wrong; look at the license for MySQL as an example of a dual license where the licensee chooses which terms to follow based on their usage.
Redistribution is unlimited for all non-commercial use. This has been through legal review.
If you’re still confused, please contact me directly and I’ll be happy to answer any other questions.
So if you do not want to allow this, then you must use something other than the AGPL. (Or maybe you could do some funky patching of the AGPL's terms, but you'd have to be careful to only do so by reference, since its text is copyrighted by the FSF, who only permit distribution of verbatim copies.)
Also, you should probably avoid calling it "fully free & open source" as you did in your original comment, since you intend for it to be neither Free nor Open Source in the sense ordinarily meant by FOSS.