No wifi in the way you were told. Access like this is tricky when the OEM doesn't explicitly support it. With my last car the manufacturer decided to basically kill the app so I went down a year long rabbit hole of replacing the functionality myself. I know this isn't exactly what you are asking for, but here are some options -
Wifi OBDII: Some OBDII dongles are Wifi based, paired with compatible software you'll see battery info. You
might be able to get enough range out of it to be useful in the way you describe. I think they are all peer-to-peer though so it's not joining your home network and proximity will matter.
AutoPi: I had terrible luck with them years ago, but they claim to support
i3 now. Last I checked they had some ICE MINI support, so MINI SE might work by combining the two? I do have the old unit still, might be worth trying again. Basically a little computer that has built in tools built to extract, record, and make remotely accessible info from the cars ODBII port. Wifi, BT, and cellular. At the end of the day you can do whatever you want to it, but you are also expected to piece together a lot of the functions yourself. If it works, I would say it comes the closest to replacing what MINI offers.
OVMS: Another 3rd party situation but the software is mediocre. Old, ugly, rigid. They do claim to support
i3 though so it may work for SE as well. Seems less flexible and harder to extend. It does exactly what it wants to do and nothing more.
DIY: I was able to sniff the commands from my old vehicle and write my own software which ran offline. From what I've seen, much of the same can be done on the SE too. I even made a sort of driving experience that read out real time speed, usage, battery, etc. It was basically a RaspberryPi, 2x CAN adapters, and HDMI displayed into the head unit. Remote actions were triggered via an app I made. I even could lock/unlock the vehicle just by walking closer or further. Huge time investment especially if you aren't already coding.