Just a quick note to everyone who runs into the problem that I faced:
Controlling your Fire TV is not possible via bluetooth or infrared, Haptique uses the “Android Debug Bridge” (ADB) interface of your Fire TV via network. Somewhere in the UI there seems to be a small hint that you need to enable this on your Fire TV (it requires enabled dev mode and then you can enable ADB access) and on my ios App I most likely overlooked that and I cannot find this anymore anywhere.
The result of a not correctly enabled ADB interface: You can easily add your Fire TV to your device pool, put it in sequences/macros and rooms but it will just not work and will not give you any error message like “ADB not working - did you enable it?”.
Also, tech support was very helpful and tried to resolve the problem but did not point me in that direction - even after I specifically asked them if I have to enable something on my Fire TV.
When I found it out myself, I told the tech support guy and he was like “of course you need to enable ADB, how should I know that you did not do that?” and sent me a screenshot of his app where the hint that you have to do this is clearly showing. Well, OK - I seem to be the first user who overlooked that hint and the fact that my question did not point them into the right direction seems a little bit odd. But hey, got it sorted out now and I just hope techsupport learns from this and does not assume anymore that a user did everything exactly as it was written on the screen or in the manual.
Some responses here, on Reddit etc. sometimes create the impression that the Haptique team is frustrated by all the frustration of their users. My message: do not give up! If someone is complaining about something and you think it is unjustified, say so but do not carry your frustration to the next guy who asks a “dumb” question or did not follow the instructions accidentially. And it does not hurt to say “sorry” or apologize if you make a mistake or found out that a bug in your software caused a lot of frustration or anger. In a lot of cultures, not apologizing for a mistake is considered rude. I say that as a German 
So, my apologies to the tech support guy for not reading the instructions. Please be rest assured that I did not do this on purpose or because I am lazy - it cost me almost four hours of my time to trouble shoot this, so I already paid for it 