I don't thing you need to cold crash it for a month. Is it fermented dry? If yes, you can let it sit under airlock for a month or so (or more) to bulk age. Pectin is insoluble in alcohol, so the pectin will drop out naturally in time.
If you want to cold crash, you could do so for about a week at most and that should clear it up pretty good. But if the cider needs age in general, better to bulk age it a month at room temp, then cold crash, rack to bottling bucket and bottle.
The haze might be malolactic fermentation, but I still am not sure if your cider was pasteurized in any way. If yes, I would doubt the malolactic. If there is a malolactic haze, meaning you are in the midst of malolactic fermentation, you absolutely do not want to bottle until it is finished.
It is possible that the cider you used had something added to it to maintain the cloudy haze of the cider. That might be the route of the issue. Some ciders put an additive in there to keep the haze, to make the cider look more authentic. But I doubt this is the case with an organic cider, but who knows...
perhaps you should describe your process in the cider making, maybe there is a clue there.