Sounds like you've all but proven the step relay is faulty. The last little bit is to be sure it isn't just the socket connections (spade terminals) making poor contact at the step relay. You could, for a test, bypass that relay just to be sure, connecting a short piece of wire between the gray and yellow.

The switch is a momentary switch. When your lights are not on, there's no sound, but the switch will give you a "flash to pass" function. With the light switch on, the step relay toggles (steps) between high and low beams. This relay has been in use for decades, but I think your 240 may use the last of them.

Do as jwalker suggested last week. Look inside. You may be able to fix it.

Headlight relay

If nothing else, its mechanical design is very interesting.


--
Art Benstein near Baltimore

The friars were behind in their belfry payments, so they opened up a small florist shop to raise funds. Since everyone liked to buy flowers from the men of God, a rival florist across town thought the competition was unfair.

He asked the good fathers to close down, but they would not. He went back and begged the friars to close. They ignored him. So, the rival florist hired Hugh MacTaggart, the roughest and most vicious thug in town to persuade them to close. Hugh beat up the friars and trashed their store, saying he'd be back if they didn't close up shop.

Terrified, they did so, thereby proving that Hugh, and only Hugh, can prevent florist friars.