While it might look frightfully low-brow, you would be better off splitting your new 12v source and putting a zenner diode between the split and each lap, otherwise you will lose the single-side parking lights with indicator and if you leave sidelights on when parked, it will power other stuff up (although your suggestion of a relay would mitigate that.
It would be easier to change the DRL bulbs for ones without the yellow phosphor, they'll probably only last a few months more anyway.
