That Amazon “EasyGuard” is pretty much what I am using for mine.
I haven’t yet updated my rfid-vehicle-unlock-and-start project. ( Where @turbo2ltr also gave me some good suggestions )
Once it is done, I will update the whole project thread because I don’t want to drag anybody down the garden path until its finished.
But for now, to help you out, Just some observations, thoughts and results that I have had.
The installation of the Push to start you linked to (mine was aliexpress but the same unit) was very simple.
As @turbo2ltr eluded to
So I installed the antenna in the windscreen and it reads the factory fob easily, I haven’t actually tested my NExT because the xEM side is T5577 for my work access, I am hopeful it will work when I get my dedicated xEM, although in saying that, I am going to move it to the wing mirror with a momentary button underneath that will be pressed by my thumb as I scan my implant (that’s the plan anyway)
Why put in a momentary switch I hear you ask 
When I travel, I do so for extended periods, I don’t want to come back to a flat battery.
(I have a hide-a-key on the outside)
alternatively, I will put in a low voltage cut out and rely on the key.
I removed my key barrel and inserted the Push to start in the cavity and it fit perfectly.
Removing the key barrel in mine removes the steering lock
My car doesn’t need a transponder, but if It did this is where you would install/mount the chip.
Now here is where it got tricky, I REALLY like the xAC, but my whole goal was to use one device to push to start and to unlock.
Yes I could have installed both and enrolled the same xEM to both…Yes I could have used 2/3 xACs, but that was not what I wanted to do.
For a few different reasons, I decided to just go with the cheap Chinese Pust to start.
The problem lies with finding a switched 12V on the board of the Push to Start when the RFID is scanned and Authenticated to unlock the doors.
The only place I could find was the piezo buzzer -ve drop, but It beeps twice in quick succession and without going into it, you can probably see the problem.
My Door locks are +vely triggered, not too much of an issue with a relay, but just one more consideration.
I am working with my mate who is an electronics engineer, who is happy to teach me some board design and programming from start to finish, It will contain a Flip Flop latching relay and will recognise the authentication peizo beep to lock / unlock.
Now for the biggest problem. TIME 
Not his priority, nor mine for that matter, but I will get it done. ( It would actually be done sooner if I wasn’t involved…) It is currently out of my car and sitting on his bench, and I won’t get back until March, nothing is happening with it anytime soon.
On the plus side
when the board is complete and tested as working, I will release all the files to the community.
and a WikiHow style update for installation.
If all goes to plan, it will be an Unlock, push to start, Lock system for under $20 ( my Aliexpress PtoS was around $10 ).
I hope this was helpful and there is some stuff to consider and not a derailment 
I was going to wait to share this in my project until I had it sorted and proven, but since you kinda asked…
I am aware there are some potential security flaws in MY system, but luckily for me, this is not something I have to worry about.
I did actually consider adding a tracking solution triggered off the same system, and THEORETICALLY, it wouldn’t be difficult, but…
There are some much smarter and more capable people on this forum than me, that can provide you further advice / suggestions…this is just MY input.
