I'm guessing you've fixed this now but the problem you have is that the new battery is not powering up the i3 systems long enough for the HV DC-DC converter to activate and take over supplying power to the 12V system.
Situation: 12v Battery failed - not able to deliver power and stay above 12v system activation threshold.
A fully charged new battery will only run the i3 for five minutes or so before dropping below the threshold for system activation. You get the red low voltage message followed a few minutes later by many, many other errors and then it enters brick mode. The REX turning on is a good thing. Its part of the recovery process.
The best approach (As documented in INSTA/D) is to get a charged battery and a battery charger that can supply 12AMPs+ . Install the fully charged battery and charger in parallel. Then you will have the i3 powered up and above the threshold for start-up indefinitely. From what I've seen the i3 consumes around 11 AMPS when powered up.
You should then find that the High Voltage controller will enable HV and then the DC to DC converter (HV ->12v) will supply power. With no charger attached you will see 14.2 volts when the DC -DC converter is active. The DC-DC converter charges the 12v battery and keeps it topped up. The 12v battery's job is to bridge the power gap between turning on the car and the DC-DC converter activating, which is usually on a few seconds when you have HV charge.
Situation: HV discharged - 12v Battery heathy but depleted and unable to supply volts above the12v threshold.
In situations where there is no HV charge but the 12v battery is healthy (but probably depleted) you can plug in the AC charging cable and this will trickle charge the 12v. Once above the 12v voltage threshold the car can be activated which should allow the HV to be charged from the AC supply. once it has above 7%(??) then the DC-DC converter will kick in.
If you have no AC available but the 12v has enough power, the REX will try and start. The REX will then try to charge the HV back to 7%(?) and supply 12v to charge the 12v battery. Nice job BMW
(1) replace the battery after 5+years and avoid all the pain ....
(2) don't ignore red 12v low voltage messages even if they flash up momentarily - (This happens in the gap between the DC-Dc activating and the 12v battery dropping below 12v)
(3) Plug in AC if in doubt.
(4) If HV is not charged and you do not have AC power available, charge the 12v and hope the REX fires up!