Schilthorn travel day advice

There are both a luggage deposit and lockers at Lauterbrunnen train station.
https://www.sbb.ch/en/travel-information/stations/find-station/station.7384.lauterbrunnen.html

There are frequent connections to Zermatt throughout the day. Every 30 minutes, at least until 8;30 pm.

I did the Schilthorn as a day trip from Bern, yes I know not ideal but it ended up working. So your itinerary is a cinch.

Watch some train vlogs that show Lauterbrunnen station so you'll be comfortable with the size and number of lockers.

Just remember, check the weather and check it again and then again.

Can't you just leave your luggage at your Lauterbrunnen hotel and then come back and pick it up before heading to Zermatt?

That's what we always do when travelling.

You have to go back through Lauterbrunnen to go to Zermatt, so leave it in Lauterbrunnen at the train station or your hotel and pick it up on your way back out. You don't pass through the Schilthorn on your way to Zermatt, it's a detour.

We leave our Lauterbrunnen air bnb April 27th

Sounds like the OP has an airbnb, not a hotel, thus the need for luggage storage.