License compatibility matters when you want to use someone elses code that was published under one license in your code base under a different license. For code that is written entirely by you, you own the copyright and can issue licenses however you wish. This includes dual-licensing.
Dual licensing is quite common: some web apps use a AGPL/commercial dual license, and the Perl interpreter is dual licensed under the GPL and Artistic License. If the two licenses were somehow compatible, there would be no need for dual licensing: you could just choose the compatible license and get the other implicitly (e.g. dual-licensing under MIT and GPL doesn't make any sense since you'd just use MIT).
To apply a dual license, I'd write a LICENSE file that offers receivers of this software a choice:
<project> - <description>
Copyright <year> <authors>
This is free software. You may use, modify, and distribute it
under the terms of either the <license-1> or the <license-2>.
<how to find the full license text>
If you accept changes to your code from other authors, you have to make sure that they are providing these changes under both of these licenses, and didn't choose one of the licenses for their contributions. For that reason, dual-licensed projects commonly include a contributor license agreement.
You may not be able to issue your intended licenses if you do not own the full copyright of the software. In particular, this could be the case if you adapt code from other projects, or if you rely on a framework of which your code is effectively a derived work. In such cases, you are bound by the licenses of the original projects or of the framework, respectively.