I wouldn't send a letter, about them 'flaunting their wealth', that is NOT my business.
If I sent any letter I would sign it.
My style would be to talk to them & ask them to consider turning them off at a reasonable hour.
Christmas time is a month at most, so this would be short lived.
One block away, a family has a wonderful display, you can even hear music when you drive by if you turn to a radio station, & it goes with the lights. Its something else & FUN!
I'm still glad he's not my neighbor.
I keep electric candles in my windows, year round, they are light sensitive & go off & on by themselves.
I'm a minimalist ;))

DO SOMETHING!