An improvement sounds good though I would say you still have some air in the system. If you can live with it then fair enough, but do keep an eye on the temperature gauge if you go on any longer runs.
Park the car so it is pointing slightly uphill. Take the cap off the reservoir and then allow the car to run up to temp at idle. When the stat opens the coolant will flow around the system and the car will burp itself at the reservoir.
In that case the fusible link may have blown. If you look on the positive terminal you should find a single wire attached. This is the positive supply to the BMS module which is attached to the negative terminal.
The fusible wire is covered with a black sheath at the positive terminal and is about 3 inches long. A continuity test will prove whether it is intact or not.
If it has blown then replace it.
If everything is working correctly then you should hear one beep on double locking the car. You should then be able to place the fob at least 1 meter away from the car and not be able to open it.
If you have no warnings on the dash and the car is running correctly then putting it on a diagnostic machine is pointless. Simply maintain the car correctly and enjoy.
You need the icarsoft LR2 at least, purely as you have the BMS on the battery. They are fabulous bits of kit though if you do get one, get it from a reputable dealer like diagnostic world.