"3 LEDs" should only require about 1.2 volts [each] at maybe 15 mils [each] which is 1/30th what you are requesting, so I am going to assume that you have 3 STRINGS of LEDs, since "3 LEDs" would look like 3 pilot lights and in my imagination would not be enough light to light up a panel...though they would do better lighting up the edge of some plexiglass.
Yeah, you should be able to use a simple circuit known as a "voltage doubler" (easily findable on GOOG images) to power up your LEDs. Your AC source is the two wires of the filament winding. You need 2 diodes, two caps. Doublers are drawn in about 6 different ways, but they are the same circuit, 2 diodes, 2 caps. It will draw (using your numbers) 2 * 45 mils = 90 mils, very little add'l load on your tranny.
LEDs are easy to use but they absolutely require a current limiting resistor in series with them. You must investigate whether whatever you have has such a C-L resistor internally, or what. Do NOT just throw 12 volts onto whatever you have without C-L, the thing will blow instantly. Hopefully, your doubler will produce something like 15 volts, you place a 66 ohm resistor in series with the supply (could be a 50 ohm, 75 ohm, etc) and drop 3 volts. The 15 volts will not blow it up, but a lack of C-L most assuredly WILL blow it up. LEDs are of course polarized, they have a + and a - and have to be wired properly to work. Wiring them backwards will not hurt them, they just won't work.
If you have a certain product you're installing, folks here can help you with this question of C-L. Do not ignore it. Post a brochure or link to mfr's literature if you have further Q's.