Switch – Light fitting won’t work with more than one bulb in place

It makes sense - look for "low voltage LED Lighting" to see what's available, or slide on over to Electronics to build your own.

Dimmable LED bulbs are those with drivers built-in that are "dimmable" - if just running controlled current to LEDs, they are all "Dimmable" (lower current, lower light) so, no, you don't want an LED described as "Dimmable" (nor one described as 12V, probably - a "bare" LED with no other parts attached is best when you are going to feed it controlled current.) If you purchase a "12V dimmer" pre-made, follow the dimmer-maker's suggestions on types of LEDs to attach to it.

LEDs are current devices, so controlling the current will give highly consistent light output (controlling the voltage, less so)

I can't find any information about the Superswitch 2701 touch dimmer but from the name alone I'm going to make this guess.

It sounds like the dimmer uses some electricity to operate. And in order to do this, is just lets some current pass through the switch for it to work. Normally, an incandescent bulb would just let this low current pass through it and not visibly light up.

That first LED bulb is bleeding off the current from the switch and it has enough current to actually cause visible light. You can either live with it, or change the dimmer to one that doesn't use any power to operate. Or you can also look for dimmers that will use the neutral if they need power to operate.