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This is a rough one. We have a low enough SPR that I don’t mind getting it in when we hit top pair second kicker, but given the nature of this board and opponent betting, it’s hard to envision the button doing this with anything that doesn’t beat us.
Since the SB has shoved and Button will need to showdown no matter what, I can’t see this being a draw, bluff, or worse Q of any kind. And even if Button is trying to force you out with a worse hand to get heads up against a weaker SB range, Button should be shoving to increase fold equity. The min raise is the worst choice with all of those holdings—I can see a flat, fold, or shove before I can reconcile a min-raise.
Finally, when we flat the SB shove, we’re doing that precisely to be able to fold in this exact spot. Otherwise, we should be shoving over the top. So this hurts, but I think you’re right, we can narrow button down to an extremely narrow value range here—this is a fold.