I've had similar trouble with a couple other "tools", (not reels) in the past, where a screw kept coming loose. I fixed it and it worked well by putting a slight "flat" on the screw thread. I did this by tapping against a hardened anvil.
Should you decide to go this route (and I think I would if you don't have to keep removing the spool) you can manage on such a short screw length using a small pair of vise grips. Tighten/fit to suit and engage the grips. Just try it a little at a time, replacing the screw and maybe tightening a bit more.
It'll work fine if you don't need to keep removing the spool. It creates an "interference fit" by flattening the crests of the threads.
Go easy to test, you don't want to mash them so as not to "start" the screw w/o the dreaded "cross-threading" trouble.
Jeremy.
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