Take a few 1.5v batteries like AA, C, D ect.. and try one led. If it lights nicely and does not burn out then 1.5V is what to use. So wire 8 leds in series for 12Vdc  or 4 leds for 6Vdc ect.. If they don't light then take one out.
 If the leds do not light at 1.5v then I connect 4 leds in series and 3 batteries, That's about 6Vs of leds on 4.5Vdc source, if works nicely I use that, if not I take out 1 led and see if that works, if not I take out another.

Nice to know that a color LED is 1.8V and white is 3V, you can add that up for close to the correct volts then go an extra led or two for safty. Then if not working take out 1 led or two if needed. 12vdc source with all color leds then should be 6.66 leds, 7 should work nicely and not fry them. Myself I would rather heve the extra light instead of wasting the power with a resister.
For all White at 3V that would be 4 leds even at 12v.

I would like to point out that as I recall the leds Christmas lights I used "SEEMED" to be a little different than normal ones. They may be the same 1.8V for colors but I think they seemed to be a little over maybe. Been a year, so not sure. This year I can't find them, I have 1/2 a string I never used yet  but where did they go?? Any way I think the ones I had worked fine at 1.5-2v each.

Also
"but LED Christmas lights are fixed, they cannot be removed and switched around like the traditional ones"

Either I read that wrong or you have something a bit different than I had. My string was exactly like normal little twinkle lights except they were leds instead. They had they little sockets, pull them out and the led leads were bent around the sides same as the twinkle lights. Just pull the leads straight and pull them out of the holder. I could put any light in any holder, mine were all color leds. No whites.
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nothing to lose