"One does not play with bears."

"Leiker" is an accepted variant in Bokmål.

"Spiller" is used in relation to games and musical instruments, so it's a less likely candidate here.

However, should you ever feel the need to warn somebody not to play chess or in a brass band with bears, then it would indeed be the correct word to use.

That's the definite singular form ("the bear"). Here, you want the indefinite plural, "bjørner".