So say after he checks his and our range are [0,1] with 0 the strongest hand, and let's say opp's calling range is [0,0.5].

If we bet a non polarized [0,0.5] and he calls with [0,0.5] our equity vs his calling range is 50%, same as if we checked behind. He does not make a mistake by folding [0.5,1] and he does not make a mistake calling with [0,0.5].

If we bet a polarized [0,0.25]U[0.75,1] then he makes a mistake by calling when we have [0,0.25] and he makes a mistake by folding when we have [0.75,1].

So just looking at possible mistakes from opp, I'd say yes, we should bet a polarized range. Seems to me the situation is very similar to that of the half street fixed limit game we looked at a while ago. We could do the same math to calculate a polarized betting range knowing that his calling range is [0,x_1]