How do you negotiate when switching shops? Asking for a friend (it’s me)

Got a competing offer from another shop. Want to use it as leverage with my current owner but don’t want to burn the bridge if it doesn’t go my way.

For the guys who’ve done this — how do you bring it up? Do you show the offer letter? Just give them a number and walk if they don’t hit it? Lay out the whole thing?

Need to figure this out by next Friday.