Is a P-trap required for a shower drain?
You need the trap, for precisely the reason you guessed.  Toilets have the trap built into the toilet itself.

What you could do is get a strainer made to fit under the plug - often used in kitchen sinks.  They are easy to find at most home stores or even dollar stores.

That will stop any big stuff from going down the drain, and give you a chance to pull it out.