Sorry for the late reply, I honestly forgot I posted here, but I figured out the issue.
Turns out it wasn't quite an issue, just a bit of a derp on my part. What happened was, I had the archery potion in my Money Trough. In the AlchemistNPC mod, If you have the Alchemist Charm item upgraded to at least tier 2, it lets you use potions from your Money Trough (Piggy Bank) with the quick buff button. The reason Auto-Renew wasn't working, is because the potion wasn't actually in my inventory, but in the Money Trough, and I normally put the buff on me by just clicking "Auto Buff". It didn't even occur to me for longer than I care to admit, that the problem likely was that the Auto Renew wasn't compatible with that feature of AlchemistNPC.
TLDR: I was being a derp, and forgot to check if the potion was even in my inventory since I usually used them from a different inventory. It works just fine when I have the potion actually in my inventory