Another possibility is the gate of the chase, shaar-ritza if you like, meaning a place with good hunting. Though gate is shaar, there are some ancient place names like Besherey (a variant of Beit-Shearim), which lend support to the idea.
It could also the gates of acceptance.
Something else to consider is that sher here means flesh, or rather, people, a variant on shear.
All in all, none of my proposals here seem that strong to me.
As for sheretz, there is the term sheritza, which meant teeming.