I think you are close to solving the problem.
Are you using LR supplied bulbs in both the signal lights and the brake/tail lights?
I almost think you are succeeding. If replacing one of the burned out rear bulbs effectively made the situation worse, then I think you are nearing a solution. Even if all the bulbs are Land Rover bulbs, consider changing all of them front and rear including those little W3W sidelight bulbs that I think you also have within the front and rear light assemblies, (in the NAS assemblies anyway).
The toughest rear signal light problem I ever had on a Chev was fixed by replacing a front signal light bulb that seemed OK. Either something internal was wrong with the bulb or just corrosion in the socket that replacement resolved, but all was well at the back after a front bulb was changed.
In the case of the 3, the computers monitor the signal light filaments, hence a signal light bulb can cause a fuss. Alternatively, pull all the bulbs out and go for a drive.
As I said, I think you are nearing a solution. By the way, did you pull the removed brake light switch apart? I bet there is back dust from deteriorating contacts within. In other words, the switch needed replacement, but you have more than one problem present.
LED bulbs perhaps not the best idea.
I know some owners replace their P21/5 bulbs with some variation of a LED and resistor, but for me, that is like wishing for an unplanned walk in the noon day sun.
I have considered replacing the P21/5 bulbs with the more common North American 1157 but so far have shied away from even that for fear of upsetting the monitoring systems. This is probably paranoia, and would solve the problem of the limited filament design life of the P21/5 bulb that is exaggerated by the fact the electrical system in the 3 runs at closer to 14 volts than 12 VDC.
The filament life of both the tail light and the brake light filaments in the P21/5 bulb is significantly less than the USA common 1157 dual filament bulb, (1000/250 vs 5000/1200 hours); brightness of the P21/5 is a bit more, (3/35 vs 3/32 candle power), but oddly, current consumption is less, (0.44/1.86 vs 0.59/2.10 amps).
It is the quest for efficiency vs maintenance trade off. It appears Land Rover regards your labour and expense to change bulbs as a worthwhile trade off in their quest for what they see as efficiency.
(candlepower / candlepower)
Too much short form I guess;probably should have written (3cp/35cp) and (0.44a / 1.86a) where cp is candlepower and a is amps.
What I intended to convey is 3 candle power brightness for the tail light filaments on either bulb and for the P21/5, 35 candlepower for the stop light filament contrasted with 32 candlepower of brightness for the brake light filament within the 1157 bulb. In reality, both are about the same, the real difference is in the amp draw. Incidentally I can find bulb specs out there that have somewhat different amp, candlepower, and hours values for the same generic bulb number. That bothers me as it tends to argue for the factory part number bulbs given that it seems there are not only manufacturing variations, but specification differences to what a given bulb may yield.
Both filaments in the P21/5 bulb draw less power than each of the filaments in the 1157. I figure the additional current draw of the 1157 would not be a problem for the conductors, but I am still afraid, (my paranoia?), of the contacts in the brake light switch.
I am not convinced that the contacts in even the replacement brake light switch are that much better designed. I wonder if even an extra short term load of about half an amp total on the brake light contacts would not adversely affect those contacts, particularly when the load is always make and break rather than sort of continuous.
