an elected Leader would be needed. the title is only a couple of words. Prime Minister sounds so authoritative so in order to establish Govt, the Leader would present her/himself to the GG and declare, as you suggest, signatories to establish a majority who are willing to work with the elected Leader.
once established the Leader could then be put to a referendum to decide the powers given to the Leader.
my personal choice would be to have a panel of experts heading each department.... but yeh I'm not sure how this would stand constitutionally.
a sortition might serve to appoint Ministers and indeed have them change annually. though I doubt this would solve the many current problems of incompetents being so appointed. far better to have people that are fully knowledgeable of each department managing the decisions making with decisions needing to be ratified by parliament before being actioned.
