I’m in the unique situation where I’m hiring my replacement after having made a promotion. I know they will make (substantially) more than me. How would I negotiate a matching or superior salary?
It's normal for managers to have reports that make more than them. Just because you manage them doesn't make you more valuable.
I'm not saying this is the case for you in particular, but it might be, so something to consider before playing hardball.
I would also say I’m more qualified for the job than those that are applying now, so I’m not sure if this really is applicable to me.
"I asked for a promotion, not for a complete career switch".
Leave or get a competing offer to use as leverage if you really like your company.
The bean counters do not give a shit about your experience or value. They just need warm bodies. They know the friction of leaving a company is high so they'll intentionally make you feel like a sucker until you finally quit, getting more years and months out of you for a lower rate than you deserve. When you finally do leave, they'll pat themselves on the back for retaining you for so long at a depressed wage.