Which is not applicable in this case. It isn’t commercial hunting, but donated by noncommercial interests. The law is simply putting an utterly unnecessary burden on society.
You speak of disease, but commercial herds get it, too (mad cow, anyone?). In some ways, the wild deer herds are less vulnerable than commercial cattle herds, because they’re fewer and farther between, and there’s less tendency for communicable diseases to be transmitted.
You also spoke of Louisiana law, but if such organizations for feeding the hungry had existed back in the day, I doubt that they would have made a fuss about it, because back then beggars couldn’t be choosers. Now many of the “poor” have smartphones and cable TV.
I strongly believe that this is bureaucracy run riot.
A couple of other notes on edit:
This place that was serving it is in no way funded by the state, and they’re not for profit, either.
And this quote from the linked article I feel sums up another facet of the real problem:
The controversy started when someone being fed at the rescue mission complained about being fed deer meat.