Ostensibly you need an Arduino that can process CANbus input to track the SoC and push that out to a cheap LTE module.
If you just want to know when it's finished you can skip the CANbus part and just monitor the current draw or something else like the battery contactor.
You could also measure the battery voltage directly - rather than decoding the report from the car - and process that through ADC to avoid the need for CANBus hardware as well.
If you have to use CANBus, you'd use a board like the one the CHAdeMO mod uses
http://store.evtv.me/proddetail.php?prod=EVTVDue2 which sold for ~$100. Their newer and more expensive one has dual CAN transceivers which you would not need for this application (but could be VERY useful if so inclined)
But since you are just looking to monitor a single bus, you could get away with a much less expensive CANBus shield (
https://store-usa.arduino.cc/products/can-bus-shield-v2) on a cheap Arduino. The cell modems would be the most expensive part at around $100 +/- $20 but with the obscene number of stupidly cheap cell phones, I would have to guess that sending SMS through a bargain basement phone over bluetooth or USB would be preferable.
So depending on what exactly you want to do, it's $200 for an Arduino with integrated CANBus and the cheaper LTE.
If you don't need CANBus integrated in one unit, an Uno and a CANBus shield are about $60 together and another 80-120 for the LTE
If you skip can CANBus monitoring altogether it's $30 for an Uno and $80-120 for the LTE modem or whatever you want to spend on a pre-paid phone. Either way, the trick is to do it as a pre-paid data plan with some cheap block of text messages to pass data back and forth. If it required a $10 a month ongoing expense there's nothing that will beat the real one. But if you can buy 1000 texts for $10 you'd have something that goes on a LONG time depending on how chatty you want it.
May not be worth it to you specifically but anyone else in the same boat, that may or may not have the option for the upgrade, should know.
If you did tie into the CANBus and had a reliable data input you could actually program the board to send any number of CANBus messages to the car. If you wanted to get really cheeky with the Dual CAN board and it allowed you to specify what the device ID was - you could imitate any other module on the CAN Bus and send messages to do anything it supported - remote start, climate control changes, etc. Read and set stuff that isn't possible even with the stock TCU hardware.
Oh, your tire light warning is on? I'll text you the PSI values reported by all the TPS sensors. Oh, you want to shut off the battery cooling but leave it plugged in to charge? Cool.