I can't speak to whether cabinet can lobby, but what I can tell you is that there are very limited matters that the minister should be considering in making his decision, and those are issues relevant to the extradition process: the treaty, the Extradition Act and the Charter of Rights and Freedoms. Extraneous issues that are relevant to the government in other contexts would not be relevant to the decision on an extradition case.
On February 4th, 2020. See this statement in context.