"Factor has new levels" error for variable I'm not using

You could try updating mod2$xlevels[["y"]] in the model object

mod2 <- glm(z~.-y, data=train, family="binomial")
mod2$xlevels[["y"]] <- union(mod2$xlevels[["y"]], levels(test$y))

predict(mod2, newdata=test, type="response")
#        5 
#0.5546394 

Another option would be to exclude (but not remove) "y" from the training data

mod2 <- glm(z~., data=train[,!colnames(train) %in% c("y")], family="binomial")
predict(mod2, newdata=test, type="response")
#        5 
#0.5546394