I feel no shame suggesting that you get the Pleco app. It is the best Character/Kanji dictionary that exists today. You can also draw them to look them up + tons of other stuff.

I only run open source software on my phone. However, I think there are some similar FOSS apps for Japanese on F-Droid, I'll have to take another look.

I wonder about this. Can't you just lock down the app to the point where you don't have to worry about it phoning home or accessing certain information? Or is it just ideology that's stopping you?

Not to my satisfaction, no. But I can't also install it in the first place because I don't have the Play store on my phone. Ideology is a part of it too.

You don't need the Play store for Pleco. You can download the apk from their website for free.