My end goal to one of my Pi3 is also eerily similar right now I have a Justboom DAC HAT installed and using OSMC
but my total end goal for that one is to actually turn it into a portable flac player (I don't have access to a battery that I could build into the Pi yet so I use an Anker 10K mAh battery its strong enough, nor do I have electrical engineering background to know what the hell I'm suppose to do if I want to use those power amplifier modules to make a built in battery for the Pi)
I was thinking about getting an Adafruit 3.5" Capacitive touch screen but my problems are how will I be able to interface with the screen with the Hat on. Justboom says I can actually solder another set of connectors from the Hat which I can stack another peripheral. But having not seen the Adafruit screen in the first place I'm not sure if I should just desolder the current connectors on the DAC HAT and use a stacking header instead I can buy one but it's online and it will be a gamble.
and then build a newer case around the entire thing and find a more miniature storage to store all the flac I could rip right now I'm using an external hard disk.
tl;dr if I use the Anker 10K mAh battery will it be able to support the DAC along with the Screen, next if desoldering the existing header on the DAC hat and placing a stacking header won't affect the HAT's connectivity with the PI3