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Material design has a bottom navigation drawer:
I would combine this with a corner floating action button to reveal it:
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In terms of User expectations, when entering into and using any Android app Users will likely expect to be able to access important information quickly and easily.
It is unclear how Users will navigate between your two main pieces of content, but it seems that by pressing the More... button the Bottom Sheet will be displayed offering navigational menu items....
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From the info you provided I see correlation with mobile games UI:
custom-made home screen with few primary actions and some secondary actions
custom-made navigation
I think you can grab the same idea.
Even if your UI is not that heavily drawn and does not look like cartoon frame, this pattern still applies and you dont need to use common navigation ...
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Short Answer
The principles for displaying hidden content that you describe work for both a dropdown and for a menu.
If you implement all of the same accessibility features for the toggle part (not the content of the 'dropdown' / 'menu' content) then you should have a pretty good accessible menu.
Long Answer / Things to consider
Without seeing the code ...
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