Pool noodles come in varying diameters. I've seen some that are like 4-5" across.
If that's still not thick enough, you could combine multiple noodles down the length and carve them into the tail shape. There are waterproof adhesives out there, though experiment first to make sure that it doesn't melt or dissolve the noodle foam.
I wouldn't shred the noodles and try to use that as a stuffing in lieu of Polyfil. You'd have to do a LOT of shredding, and get it down into really tiny pieces to avoid the tail looking lumpy. It would be way easier and less time-consuming to just carve down larger pieces into the tail shape.
Even if you sewed Polyfil in a waterproof fabric, there would still be a seam for water to leak through, so you'd also have to totally cover and seal the seam and all stitches.
I don't know how latex would react to pool chemicals. I've never exposed it to chlorine or extended water saturation before to know whether it'll be fine or if it'll cause it to break down, absorb water, etc.
You could try looking online for merfolk communities - these are people who are really into mermaid/merman costuming. It'd probably give you a lot more ideas about how to make a swimmable costume, what materials are pool-safe, etc.
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