Playing the iOS/Android ports of Sonic 1 and Sonic 2 on a TV

[BomberDino]

Here's what I have:
iPad 4 w/ HDMI adapter
Note 8
Xbox 360 controller
Generic USB controller

I have an iPad 4 with the Retro Engine remastered Sonic games installed. When I connect it via an HDMI cable to a TV, I get lag. Has anyone done this with a newer device? I was thinking about purchasing a 4th generation AppleTV, but those are still expensive on eBay.

I don't think the Xbox 360 wireless controller is MFi compatible. Is there any MFi-to-USB bridge I could use with any generic USB controller?

Alternatively, I could purchase an HDMI adapter for the Note 8 and try with Android. Has anyone whose gone this route experienced issues with lag? I would also need a USB adapter to connect the generic controller, but I don't think I can connect that at the same time as the HDMI adapter.

Maybe there's an option I'm missing here. I know I'll need to purchase extras. I just want to keep it as affordable as possible.

Also, is there any way to disable the two onscreen circles that are used as a controller?

[Ziggy]

I have Sonic 1 on my Fire TV (not the stick) and I played it with a Bluetooth game controller. It's been a few years so I don't exactly remember, but there was definitely something that felt off to me about it. I don't know if it was lag, or maybe because it was widescreen, but it just didn't feel right and I couldn't play it.

[BomberDino]

I have a 1st generation Fire TV Stick, so I can't use a USB controller. However, I should be able to pair a Bluetooth controller with it. I haven't used the stick in over 5 years. I hope it still works with the recent updates to the game. It's worth a try, just have to borrow a Bluetooth controller from someone. Thanks! (Thinking it would make a decent travel gaming system if I can find other controller-based games for it.)

If that feels off to me, I'll just stick with Sonic Jam or Sonic Mega Collection for classic Sonic on the TV.

[Anapan]

I don't know if a 1st gen Fire TV stick would work, but at least later models would work with a "Charge and OTG" cable where it gave you two female charging ports, one for external power, and one for a secondary device (some hubs worked). People used them mostly to inject ethernet into Android boxes that were wifi-only due to lack of ports, but they were fully USB compliant. This might be a solution.
edit: sorry, possibly it can be enabled with root but "Make sure you have 2nd gen Amazon Fire stick, OTG function not supported on 1st Gen Amazon fire sticks."
I have some old unused android boxes better than a 1st-gen amazon stick. PM me in the next couple days and I'll mod and send you one before I leave for work. I hate to leave these capable devices collect dust, but they're not worth anything. I'll throw some generic OTG stuff so you can do the things.