Properties of potassium carbonate as leavening agent We have a gingerbread recipe calling for potassium carbonate (vulgo baking potash). One year I replaced the baking potash with baking soda 1:1. It worked fine.

Later I researched what the supposed differences are and found a claim that you may leave the dough up to several days in a cool place to rest when using baking potash but not when using baking soda. The (family) recipe originally also allowed to substitute baking potash for hartshorn salt (ammonium carbonate), but since this has the effect that the dough gets a little runny in the oven and therefore the resulting gingerbread is flatter than desired, we haven't used this in decades.

This answer describes potash as "purely historical leavener". Alas, where I come from (Germany) it is commonly used in Christmas recipes. In fact you will find it alongside the respective spices during the season.

The recipe of my family explicitly allows for storing the ready dough in a cool place for several days. Another recipe of the family of my significant other actually calls for leaving the dough in a cool place for several days and also uses baking potash. I never thought anything of it, but "the internet" appears to ascribe preservative properties of sorts to baking potash.

Or is there something else behind the alleged difference in keeping the dough in a cool place? For example, is baking soda perhaps known to somehow dissolve or otherwise react with the other ingredients and thereby lose its leavening effect?

Could someone please shed some light on the properties baking potash would have in a recipe aside from the leavening effect?! Please provide authoritative sources, if possible.
I've never heard of any preservative effect of potash, the reason it is still used in traditional recipes is the ashy flavor it imparts. That smokiness is not to everyone's taste, baking soda is a perfectly good substitute if you prefer the end result.  

You can refrigerate baking soda leavened cookie dough for 3-5 days without issue, it's batters where this would be a problem, but it would be whether you used potash or baking soda. The reason is that any chemical leavening agent starts to react with acid right away and produce carbon dioxide. Refrigeration slows this process, but doesn't stop it - cookies don't need as much leavening so it's not an issue whereas for a cake it's a problem. 

Refrigerating a spiced dough for a couple of days is a good step because it lets the flavor of the spices come out and blend. You can achieve the same thing by mixing the spices with the butter and refrigerating the butter for a couple of days instead, the end result is the same from a flavor point of view.