Should I use flash with LED lighting?

there is no right or wrong answer to this question, photography is an art form and like any art form what makes a good photo in one persons eye is different to another's. The cameras flash will most likely wash out the blue light from the LEDs in your room as no doubt the cameras flash will be more powerful, you will get the blue colour showing on then photo, but not to the same scale as on your phone. I'm afraid you are right in thinking this will be a trail and error thing, you could experiment with the settings available on your camera. you can even use different colour plastics placed over you flash to give similar effects.

you can use the white balance on DSLRs to give the kind of image I think you are after, but wouldn't know how to do this on a polaroid instant camera.