I have same problem with Red Samurai, I'm trying to use Genesis/N64/Snex/PS1 Emulators and without being able to reprogram Start I'm lost. does the Nyko Propad Plus, Logitech F710, or PS4 Dualshock have this issue? and which of the 3 will work will all emulators. Trying to find a Kid friendly solution that once it's programmed it'll work consistently.
Honestly if you wanting a kid friendly dedicated emulator machine then get an Ouya. Supports multiple controllers out of the box and pairs easily with PS3 and Xbox and all Bluetooth controllers. The ouya is hands down the best gaming box I've found that just works.
The fire TV stick is great for travel but picky on getting everything to work. Seems some apps work with certain gamepads and others work with others. I loved the feel and ease of pairing the Red Samauri. I might keep it around for my Ouya.
I just bought a Nyko Playpad Pro but haven't tested it on the Fire TV yet. Worked fine on my Nexus 7 with Snesx. I get a minimal response delay on both the Nyko and Samauri in super Mario world. Mario games are great for testing delay times. I don't experience this on the Ouya.
Update
Just tested my Nkyo Pro and it works great. I didnt fully test everything but terraria works as expected with the trigger issue I had with the Samsuri.
Overall I like the Samuri feel better... reminds me of a PS3. Felt great in hand and in game. The Nyko feels like a 360 controller and kind of clunky. Feedback on the buttons arent the best. Seems I have to push down pretty hard to get the buttons to register. They may losen up over time. The joysticks feels good on both.
Mouse mode works great on the Nyko. Great option to have on the fire stick for traveling and web browsing.
The Samsuri has keyboard mode and Gamepad mode. I had to actually pair using both modes for it to work. The difference is keyboard mode would work better for mapping probably as each button is assigned a letter off a keyboard. Up = W vs Up = Up in gamepad mode. Might possibly be the way you need to setup 2 controllers; 1 set to G and ths other set to K.