I would point to New Zealand and the United Kingdom, both of which developed cabinet manuals. The idea was not to change the conventional rules, but to at least codify them in writing, so that the public and parliamentarians could see and understand them.
That approach does not involve changing the Constitution; the manual merely provides guidance in the form of a non-legal document. We are talking about a manual that is agreed to across the board and debated before being approved. Accordingly, it serves as a beacon of sorts, helping to interpret these very important rules in the midst of a crisis.
The committee heard from my colleague Hugo Cyr, who made a case for strengthening the rules by holding a vote beforehand. Those are options you could adapt and fine-tune through the usual parliamentary procedure, including committees. Ultimately, though, Parliament's ability to undertake reforms is limited.