Try it. The AI will probably tell you that it's, of course, doable. You would have to start by making your own AOSP distribution and require an unlocked bootloader to even attempt to install it. You definitely can throw an AI agent at the problem, but a) it'll be significantly more than $200, no matter how you cut it; b) you'll end up with tens to hundreds of kloc of AI-generated code in a security-conscious context; and c) you can forget about having more than a handful of the most desperate users[1]. Both b) and c) are fatal for a project like this.
The locking down of the Android platform, IME, is a massive, decade-long process[2] with "full speed ahead" corporate backing. Even just a few years ago, you could maybe code around some of the restrictions (if supported by users going into settings and tapping some checkboxes); today it's impossible even with root. To get working "push notifications" outside of the official channel, you need to hack the support into the OS - or accept that you probably will get the notification, but it can be anytime from a few minutes to a quarter hour before your app receives it.
[1] In which case, making them use tens of thousands of AI-generated code "for security" is a clear moral hazard you probably don't want to walk into.
[2] I don't want to judge whether it's a move in the right direction or not - that's a separate matter. But it is happening.