Hey I know this feeling. It helps to think in terms of "who's paying the bill". There is simply no way you can gain EV by making (or not making) a zero-EV bet. If you still do, either the bet is not zero-EV - or you won't gain that advantage :laugh:
A related question would be, if you cannot make both plays (for whatever reason), which one to chose for less variance if both are zero-EV ?
Variance Surrender: 1/3 * (-1)² + 2/3*(-.5)² + 4/9 = 10/9
Variance Insurance: 1/3* (0)² + 2/3*(-.5 + X)² + 4/9 > 10/9
Here, X is the hand after Dealer peeks. Since surrender is zero-EV, EV(X) = -.5. For X identical to -.5 (neglecting variance on X), one gets the same 10/9. As X does fluctuate, variance on insurance decision will be higher.
So, if both bets have zero-EV, and you cannot (or don't want to play insurance/surrender), you should surrender.