I do think you will need to go even further because as you can see in developer options, on your capture that only adb connections is avaiable to be altered, you need to be able to "allow unknown sources."Yes the newer Fire Stick lite don't have the Developer's menu activated by default as its prior version did. They now work just like a regular android phone. To activate the Developer's menu you go to about and click several times on the device's name (Fire Stick Lite in this case) until a message pops up notifying that the Developer's menu is activated.
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Then you get this:
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After this you can proceed with installation as usual. If this doesn't work for some reason then you'll need to access the hidden developer's tool menu which is a much more complicated affair. A google search with the following term: Fire Stick developer tool hidden menu, should get you a couple of tutorials on how to activated it.
Had to redo post, I left personal info on one of the screen shots.
I just googled how to get to the menu, but I don't know what to do once you're in it being as though i don't have a firestick.
Remote Control: If you will be using a remote, then press and hold buttons DOWN and CENTER for five seconds. Then, release the buttons. After that, press the button MENU.
Once youre in, look around for something to enable unknown sources. once you enable the right thing, then unknown sources should appear somewhere in the developer options.