What kind of flour keeps bread dough from sticking in a proofing basket?
Regular old flour works -bread or all-purose or even cake- but the dough had best be proofed away from too much moisture, ie no steam. Use an overly generous amount sifted onto the basket and with each success reduce a bit til you find a happy place.

Even spongy rye masses have come out in one piece with a jiggle-jiggle here and a hop hop tip. Think of that as walking the dough out instead of inverting.

Bits may stick but only minor flesh-wounds. Let basket dry and scrape clean.

If the problem is related to brand-new basket, may need to search for some 'seasoning' tips first.