Print with larger holes, but then use multiple layers. Like Russian nesting dolls, but each "doll" would be rotated very slightly.
Also, try slots in opposite directions. Horizontal slots nested inside vertical slots would effectively create a mesh.
Flat panel inserts could be printed easily and then slid into a circular holder.

What if you printed it flat as a rectangle and then rolled it up and sealed the edges? Seems like it'd print better that way, no retractions, no bridging, no fragile areas of like ten 1mm-width layers stacked on top each other.

Ditching the PETG would probably go a long way. Try it in ABS or even PLA and you'll have better shape definition and a whole lot less stringing.

Rather than reproducing the pattern in the original wholesale, you could also consider stealing the idea from e.g. this, which has a "mesh" outer wall that's a continuous loop and thus doesn't require a zillion retractions every time there's a gap.

Try a smaller nozzle and dry that filament till it grows a mouth and asks you for a glass of water.

That's a lot of retractions. What is this filtering? Maybe you could print a more open structure and line it in mesh?

Also, where is this part in relation to chlorine? I can't imagine a printed part would hold up very well in high chlorine concentrations, but PETG is fairly inert...

~~Tangent: That sounds like a bad idea for food safety. And I'm referring to both the original and the recreation. If it were for myself, I would buy something made of metal.~~ edit: Thought it was a coffee thing, disregard.

On topic: Could you clean up the holes with a hot needle and some patience? PETG tends to be very stringy, which is probably the reason the holes are not well defined. Maybe try tuning your printer to minimise stringing?

i would try printing the filter part as a flat surface (without retractions) directly on the printer bed (only 3 layers maybe) and bend it later to become a tube ( like a "C") i am not sure about petg, i did this bending with pla. hot water comes in handy and maybe a glas bottle. also prepare something to tighten where the ends of the C connect, a border without the small filter holes but some larger holes for something like cable ties or even a printed part for connecting the C edges.

U van probably get some improvements by dialing in the printer settings. Otherwise just change the geometry would probably be more effective for less effort. I would reccomment circular holes with the same area as the existing filter would help reduce the issues ur seeing with overhangs. And maybe turn the cooling fan up a touch to help reduce stringing but it probs wont be too effective considering the material.