There’s a good reason the manual recommends not towing in the break in period, and it is primarily related to the differential. Heat is part of it but really a by-product of the break in process. However, it also aggravates the conditions during break in by thinning the oil.
When the differential ring and pinion are new, the gear tooth contact area is very small, creating very high contact pressure. This causes oil film breakdown and surface wear, which eventually wears in the contact point, spreading the tooth contact pressure and reducing the local surface stress. If the differential is highly loaded before this wearing in has been allowed to slowly occur, the high local contact stress can cause spalling of the tooth surfaces at the contact point.
If spalling occurs, best case it will cause a noisy differential with reduced life, and worst case it can cause tooth failure due to the created stress concentrations.
instead, you want to run the differential lightly loaded until the gear contact patches have worn in to some extent, spreading out the force and reducing the contact pressure. Once this has occurred to some extent, the gear teeth can withstand the higher load assocIated with towing. GM has recommended a particular minimum number of miles for this. More is better.