One final thought from me and then it’s over to the auto electrician.
I am wondering exactly where the new relay is physically positioned. If it is behind the dash and close to the speedo itself or, in close proximity to the speedo pulse wiring, could it be an electro-magnetic pulse is induced into the speedo or wiring at the moment of switch on? Could be worth mounting the relay on some extended leads so that it is well away from the loom and instruments to see if problem persists?