The sugar helps with browning but can detract from a crisp exterior. The sugar you use for browning can be greatly reduced or even eliminated with a longer (thus colder) ferment; thus, you have a better chance at dough that browns as well but has more of a crunch on the outside.
You can certainly follow your plan with same dough at room temp vs. cold ferment, but the length of time and temp factors into how much yeast is optimal that they might not truly be "equal" outside of those factors. Eventually, I'd recommend keeping all your non-yeast ingredients the same but changing your yeast % for the cold vs. room temp ferments.
Below is the chart I use to figure out yeast, temp, and fermentation times. These aren't to be taken rigidly but are seen as a general guideline.
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