I think it was probably accidental. Peter Fonda bought the watch to celebrate having written the screenplay, as it was his first. He probably tossed the Timex because he didn’t want to throw his new Rolex, and as filming is rarely done chronologically, he just forgot and had it on in some scenes.
You’re dead on that it was beatnik / hippie inspired, I’m just not sure Hopper (director) and Fonda (writer, producer) were mature enough filmmakers to be thinking about the watch beyond throwing it down.
Just my theory. I’m probably wrong. We will never know because they’re all dead.