Unlike a burner phone, you wouldn’t constantly be throwing it away (just getting new SIMs every once in a while) so cheapness would not be the #1 priority—but it would still be #2 or #3, since this is still a whole 'nother phone you have in addition to the phone you usually use. And you’d want it to be ergonomic to text on, so a 1990s candybar dumbphone wouldn’t quite cut it. It wouldn’t need apps like a modern smartphone (other than maybe messaging apps.) And battery life would be important—it's often hard to find private time to charge a secret phone—so a big screen would be a drawback, not a positive.
Ultimately, I figured that an old Blackberry would be perfect for this use-case. Guess I deduced correctly.