I really don't think it's the switch.
If you do a resistance check, that's good enough.
Position 1 should be infinite resistance in one direction and a few ohms in the other(lug 1 to 2 and 2 to 3)
Position 2, the same but reversed.
The only way I could see it being the switch was if in the LED position, the internal mechanism contacts have a near intermittent connection. This could be tested by puting light stress on the switch, jiggling the switch lever and thumping/vibrating the switch. Something that would cause a shoddy internal connection to possibly open.
Also try jumpering the relative pins to see it it resolves the gating(1+2 or 2+3, whichever it is). Aligator clips or tack solder a resistor leg across them.
I'd pull the cap and stick a big ass film in there before desoldering the switch.