Does the case of strategic complements always result in a situation where the Stackelberg leader is worse off?
It depends on what you assume the firms know about each other.  Under full information / rationality assumptions, with identical firms, second-mover advantage disappears because the first-mover will recognize the credible threat of undercutting, and set P = MC and that will be that.

If the firms are not identical (e.g., different marginal costs) then turn order is irrelevant, because only one firm has the ability to make a credible threat of undercutting.  That firm will capture the market, whether it moves first or second.

Only if you drop the assumption of full information / rationality is it feasible that turn order would matter for Bertrand competition, because then the assessment of credible threat becomes a probabilistic one.