I personally haven’t seen that info message before, but hopefully somebody else has that can confirm my ramblings below.
The following is just my best guess 
TagWriter, TagInfo, NFC tools etc only “talk” to the HF side of you NExT.
Obviously you have tried to enroll it, which is probably the best way assuming it is the same mode as your system,but, either way it looks like your LF side T5577 chips UID is outside of your systems parameters
Your NExT T5577 chip comes Pre-programmed in EM41xx mode with a 40 bit unique ID, so I am not sure if your system is running EM but your UID is outside what it is looking for or if the system is actually running in HID, Indala ProxCard etc… mode.
(you could maybe do some more fault finding, like getting more info on the system, scanning your authorised card to see the UID, scanning your NExT on a reader to see if it at least shows its UID (maybe what you tried and got the info message)
Either way, unless your system has a writer that you could write to your NExT, which I highly doubt (it probably is just a reader that reads a preconfigured new card/fob to then enroll it in the system)
So I think 
: your best bet from here is likely a Proxmark (You might get away with a Blue cloner, but the Proxmark is probably the better option ) And the ProxMark3 Easy is a happy medium between the RDV4 and Blue cloner.
Definitely doable, but sounds to me at least, unfortunately you have one last hurdle.
Keep us in the loop, I hope that all makes sense.
Ask again if you need clarification.
Also I’m happy for somebody else to chip in if you have another suggestion that may be easier