Don't those prereqs all do something like "3rd+ level spells"?
No, they don't. Otherwise, we wouldn't be having this conversation.
I would say it depends on how you get the spells. If it is like a wizard where you may learn a lower level spell in place of of a higher level spell you would only need to learn one of your 4th level spells as a 3rd level choice instead. You still wouldn't have the slot but could use a 4th level slot to cast it. You know have a 3rd level spell that should qualify you.
Yes, but we're talking about a
Nar Demonbinder. They have a fixed spell list, which doesn't include
anything below 4th level. Nar Demonbinder was quite popular right after the UE book came out, and I saw a lot of builds jumping through hoops to use it to qualify for other PrCs (and made some myself). Arcane Disciple and Mother Cyst were the easiest, IIRC.
If it were a Sublime Chord, then yeah choosing a lower level spell known might work. (I forget what it takes to qualify for that PrC, though. You might be required to have 3rd level spells already, or something...)