No don't include the response variable, just include the vectors of predictors.

There are some models where you would pass the entire matrix and then specify which is the target, but in general unless you are sure about that you don't want the response included as a variable when you train the model. In this case, since you are passing a vector you can remove that column from the matrix.

If you haven't come across it you could check out Deep Learning and Scientific Computing with R torch by Sigrid Keydana. It's free if you google it. Chapter 13.2 contains an example dataloader using Palmer Penguins.