I've used the nylon paint strainers many times. The best thing about them is that they are reusable! You can definitely use it for bottling, but make sure you soak it in Star-San or some other type of sanitizer for a couple of minutes first. I used to filter my beer before bottling, but after awhile it became a PITA! So I racked into my bottling bucket, sanitized my SS spoon, and made a whirlpool after adding the priming sugar. This helped a lot with keeping sediment out of bottles.
I doubt a strainer bag will help much at this point. They are probably better used as hop sacks or cheap BIAB sacks or straining the kettle to fermenter.
Once fermentation is complete all the stuff will have compacted to the bottom. Cold crashing helps even more. Any sorta "haze" that you are trying to filter out will likely pass right thru the mesh. it just isn't that "fine" of a filter.
Maybe mix in some gelatin in the bottling bucket maybe? Just remember to decant the bottles carefully when drinking. But not sure about that since I haven't bottled in 25 years.
I use a bag to keep pellet hops out of the bottling bucket when I have dry hopped. I put the bag over the end of the siphon tube that I put in the fermenter so the hop particles stay in the fermenter and don't get in the bottling bucket.
For many years I didn't use a bag, and still mostly kept the hops out of the bottling bucket, but a few got in there and would end up in the last couple of bottles. The hop particles act as nucleation points if they get in the bottles, and can cause the beer to gush out when you open the bottle.