Lamentation is it’s own multiplier, so you can just straight up compare the multipliers between two belts without having to worry about anything else.
You just take the ratio of (1 + decimal version of Lamentation multiplier). The 1+ is there since it’s a bonus on top of your regular damage. E.g. if your regular damage is 1 and you deal 50% extra damage, you then deal 1.5 damage (1+50/100). For a 150% Lamentation, if you deal 1 damage before factoring in the belt multiplier, you deal (1 + 150/100) = 2.5 damage after.
I think you had the right idea, but you were missing the 1+ for your baseline damage in your calculation.
So to compare a 150% belt to a 113% belt, a 150% belt is (1+150/100)/(1+113/100) = 17.4% more damage than a 113% belt. A non-ancient belt with 140+ bonus will handily beat an otherwise well-rolled ancient 113 belt in the damage department.