I agree with the advice to reduce the amount of water in the recipe. Be sure to account for ALL of the water added to the batter -- including water in any additives and colorants.
It's really not necessary to avoid gel or to go to great lengths to keep the soap cold, however.
The reason why "rivers" happen is the combination of two things -- the soap going into that liquidy gel stage AND THEN the slow cooling of the soap batter. If the soap doesn't go through both things, "rivers" are unlikely.
Colorants, titanium dioxide especially, enhance the appearance of "rivers" but they don't cause them to form. I've seen "rivers" in uncolored soap, although the streaking and mottling are much less obvious than if I'd used colorants in the soap.
You can prevent the soap from going into full-blown gel by reducing the water content. Less water increases the temp at which the soap will gel. If that's all you do, that should be enough to prevent rivers without having to jump through a lot of hoops to also keep the soap cold. Most of the soapers who soap with about 33% lye concentration (I mean lye concentration, which is not "water as % of oils") don't see much if any "rivers." At lye concentrations of 31% or below, rivers are more likely.
But if you want extra insurance, just set your mold on a cookie cooling rack or put it on some cans of soup -- anything to get some air underneath the mold -- and put a fan so it blows on the mold. That will help the soap cool down quicker and further reduce the chances that "rivers" will form.
More:
https://classicbells.com/soap/streakMottle.html