Focus on the character's experience.
Your narrator feels distant from your character and that's why you're struggling with word choice (and I realize you are only giving us short bits from your narrative here). Get in there and tell the reader about the character's emotional state.
He heard a strange noise. Was he curious or freaked out that it sounded like a heartbeat? Did it sound like a human heart or an animal's? (speed is a factor here) What did he imagine was making the noise? A baby? A wounded deer?
He found the object. What did he see and how did he feel about it? Was he surprised?
He picks it up and feels the beating directly. "Throbbing" is a good word here because it doesn't apply to what you hear but to what you feel. Did he almost drop it out of shock? What was the sensation on his skin? Did it travel up his arm?
"Beat" and "pulse" refer to rhythm and you're right that both can work for what we hear and what we feel. You want the reader to have that sense of timing in the background during the story (which you can do with the prose itself in addition to using these and other words to set the stage and periodically remind the reader). But you need to show the reader what the character is feeling. Literally feeling in his ears and on his skin and also emotionally (if your narrator isn't one that get's into a character's head, you can still show the character's reactions which will have a similar effect).
How someone reacts to sound is different from how they react to touch. Even if the same underlying rhythm and the evoking of a heart is present in both cases. By focusing on the character's experience, you won't be relying on the rhythm words to carry your narrative.