I really like your flop sizing.

Assume [a worst-case scenarios where] his calling range is all sets. Against sets our equity is something like 2/9. We're shoving about 10.5 into 13, the final pot will be about 34, so we're getting about 68/9 back, which is like 7.5 or so. Since we're betting about 10.5 but getting about 7.5 back, we're only risking $3, and 3/(3+13) = 3/16 = 0.1875, so we need him to fold about 19-20% to break even on a shove in terms of absolute EV after the rake. It's anyone's guess how much we need him to fold to make it better than checking because checking is such a complex scenario with too many variables we can't estimate well, but it's probably somewhere in the 22-26% range.