Messages telling your 14-year-old to jump off a bridge and kill herself aren't "teenage tiff language" by any stretch - that's a serious safeguarding concern, and this cannot be left, brushed aside, or quietly smoothed over. Your instinct that doing nothing isn't an option is completely right.
This needs formal attention, and honestly, the police should be involved. A message actively encouraging a child to kill herself is not a grey area - it's the kind of thing that can constitute a criminal offence under malicious communications law, and it needs to be treated with that level of seriousness, not handled quietly between families over a cup of tea. Whatever this boy's reasons, there is no version of "he was just upset" that makes telling a 14-year-old to jump off a bridge acceptable. That's not normal teenage fallout, that's genuinely alarming behaviour that needs a proper, formal response.
Your daughter not wanting police involved is understandable, but this may not be solely her call.
Teenagers often fear the social fallout of police being involved more than they fear what's actually happened to them. But as her mum, when a message like this is sent to your child, you are entitled - arguably obligated - to take it further than she's comfortable with. She doesn't have to be the one who "gets him in trouble"; you can frame it as you, the parent, reporting something you found unacceptable and dangerous.
The school also needs to know, regardless of the police route.
They have a safeguarding duty, and if this boy is capable of sending messages like this, the school needs eyes on the situation for your daughter's protection and for anyone else he might target next.
Going to see his parents can still happen, but as one part of a bigger response, not instead of the serious steps.
Screenshots in hand, calm but absolutely clear that this isn't teenage drama, it's a child telling another child to kill herself, and that alone warrants far more than an awkward doorstep chat.
You are not overreacting. Somebody sent a vulnerable 14-year-old messages telling her to end her life, and that deserves formal, serious attention - police, school, the lot. Trust that fury, it's entirely proportionate to what's actually happened here.