It is not possible to give a definitive answer if your plan complies with the GPL license restrictions or not.
Your Program2 can be seen as a plugin for Program1, and the GPL FAQ has this to say about plugins:
When is a program and its plug-ins considered a single combined program?
It depends on how the main program invokes its plug-ins. If the main program ...
No, you don't.
The AGPL Section 13 only requires you to publish the sourcecode of an AGPL-licensed program if you "modify the program". When you host an AGPL-licensed software as-is, then you have no further obligations.
And even if you were modifying min.io, that provision only applies to "users interacting with it remotely through a computer ...