I don't usually suggest this but I think you should consider using Tri-Chlor pucks for a while. Get a couple of new floaters, and a bucket of good Tri-Chlor pucks. Avoid "Double-acting" or "Triple-Acting", and avoid ones that contain copper (HTH's double-acting).

You just want plain tri-chlor. Buy 2 or 3 boxes of 20 Mule Team Borax too.

Here's what I'm thinking:

Your CYA tests at zero and your FC see-saws. Your pH is OK and your water is clear, no obvious algae

Use the tabs at a continuous erosion rate and your FC should be very constant. Meanwhile, the Tri-Chlor will be slowly dumping CYA into your water. CYA's job is to prevent too-rapid depletion of chlorine, but you are taking care of that with the constant erosion of the tabs.

Next you'll need to watch your pH because the Tabs are VERY acid and will drive it down. That's what the Borax is for.

Floaters are cheap--adjustable ones are $10-$15.
An inline-chlorinator is far more expensive and not necessary.

For something to be "eating" your CYA, you have to let your pool turn completely into an algae pond and let it stay that way for a month or two--YECH!!! Either you don't yet have enough for a reading (I THINK you need 15 lbs for your pool--but don't take that as gospel), or you are testing incorrectly, or you didn't add as much as you thought, or you've replaced more water than you realize.

Now, instead, of this, you can keep adding CYA and using bleach to chlorinate. Absolutely nothing wrong with that.

I don't object to Tri-Chlor tablets IF THEY ARE USED APPROPRIATELY! This would certainly be an appropriate use of them.

Good luck!