Another thing might be useful. At Westminster they have the committee of chairs, the liaison committee, the backbench committee and other vehicles that are not yet established here and may never be established here.
Since, for example, you have a subcommittee that deals with private members' business, if you really wanted to be bold, you could take on the responsibility or suggest that you, as the procedure committee, would be prepared to assist in setting the agenda of the parallel chamber.
I'm just throwing that out there for your consideration.