I do not know whether the Senate Speaker has the same access to the Governor General. I can't see a reason in principle why not; I've heard of no convention to the contrary.
If the advice, and here I don't want to misspeak.... It's not so terribly important if the information is contrary from the Speaker of the House or the Speaker of the Senate. Because it's not advice in the constitutional sense, as it is from the Prime Minister: this is just some background information saying that these are things you need to be aware of when you are making your decision.
I would be very surprised if it were not already the case that the Governor General has multiple advisers who do not always agree.