When you say you "cross the two legs", do you mean that you connect them together?
If so, I think that you may be correct that the LED itself it blown.
Have you got a multimeter that reads current? I guess that is one that can be seen in the first picture. Can you use the multimeter, set to read current, to link the two legs of the LED and make the others light? When you do that, what is the current?
The choice of LED isn't critical. You need the right colour, but most likely any red LED will do. BWM will match the colours very carefully, to make sure that it is exactly the correct shade in the showroom. If it looks red to the drivers behind, and you are happy with it, that's all that's needed.
You need an LED that fits in place. I think that is a 3mm LED. You measure the diameter on main part, not the wider flange. If you can fit a larger LED, that's fine. The individual LEDs can't be seen when the car is in use, so it doesn't matter if they aren't the same size as the original. Obviously BMW will have one size for mass production.
You need an LED that can take the current that the circuit provides, or more. If you measure 35 mA, then any LED that is rated at 35 mA or more is fine. A 50 mA or 100 mA LED will be work. LEDs will degrade quickly if driven at too much current, but they are actually more efficient at lower currents than their rating. There is only a problem at tiny current, like less than 1% of their rating, but that won't happen for you as you don't have enough space for an LED with a far larger rating than needed.
You don't need to worry about the working voltage of the LED. It will be around 2 V for a red LED, and the control circuit will take care of any variation.
It's a good idea to get an LED with a similar viewing angle to the ones you have. Just look and see how close to straight on you have to have the axix of the LEDs to your eye to be bright. Search for "led viewing angle" if you're not sure, but it's not an exact measure, and again you can be less fussy than BMW will be, and if the rear light looks OK, that's all that's needed. You do need the rear lights of the car to be a similar brightness when the viewer is off to one side of the rear of the car.
You need to fit the LED the correct way round. The circuit has two LEDs in series, the working ones, with one driver circuit and three LEDs in series with a separate driver circuit. You can see the top of the pins, inside the clear plastic, and the larger one of those is negative. You can test the polarity with the multimeter on continuity or a low Ohms setting. Red LEDs usually light up dimly when tested the right way round with a multimeter.
I guess that the LED that blew was the top one, that ran hottest. Getting larger LEDs will help stop that happening again. You might change all the LEDs, as the others may be degraded.
It's a good idea to compare the brightness of the new LEDs with old at some point. LEDs do vary in efficiency, and if you get it badly wrong, the new LEDs could be too dim, or less likely, too bright. You don't want the car tail lights to be nearly as bright as the brake lights.
You can get LEDs from Farnell, RS, Mouser or Digikey. They can be searched by diameter, colour, viewing angle etc.
It's quite likely that other BMW lights use the same LEDs, so you could salvage from them. You could just salvage LEDs from both sets of lights to get one working light and then buy a new light for the other side. I did something similar with a BMW Mini, where the springs on the door check strap rust out, but only on the top where the water can't drain. I bought a new check strap for one side, and used the bottom spring to replace the top on the other side.