What you are saying is that if you ignore all the blinking lights and the like, your 3 is now routinely both starting up and running - for the time being anyway. I of course, share your sense of foreboding.
As such, you might try what is called a hard reset. The purpose of that is to remove/clear/erase some of the trouble codes that have built up in the various recording systems. One hears stories of the systems being so full of codes from a previous fault that even if the vehicle has been fixed, the trouble sensing systems keeps tripping over the old codes. That is why dealers will clear fault codes after "fixing" a problem.
Unfortunately, not all codes can be cleared via a hard reset - many codes can only be cleared with the T4 or a Faultmate however it costs nothing to try the procedure and sometimes it works.
Also if you have a volt meter, I would keep a watch on the battery voltage across the + and - terminals of your new battery when the 3 is shut down - and particularly in the morning before you start up just to see if voltage is being drained overnight.
In brief, to start the hard reset process, open the hood but close all doors and let your 3 go to sleep - no radio display etc.
It is often a good idea to have a window rolled down and the key in your pocket whenever fooling about the vehicle.
Disconnect the ground battery terminal from the battery.
Disconnect the positive battery terminal from the battery.
Connect the negative cable to the positive cable, (NOT the battery). You will need a short length, (a foot or so), of light gauge, (14 to 18 gauge, AWG, stranded copper), wire to do this as there is not enough slack in the positive and negative battery cables for them to touch each other.
Hold them together for about a minute or so, (some say 5 to 10 minutes); you are discharging memory modules within the engine computer and elsewhere.
Reattach positive terminal to the battery.
Reattach negative terminal to the battery.
Start engine and hopefully no warning lights.
Reset time on the radio - note that the station presets are still there as not everything is erased. Nothing much else needs resetting either.
This is a link to a thread in disco3 re hard reset.
DISCO3.CO.UK - View topic - Hard Reset