If a private members' bill involves the spending of public money, that bill can survive until third reading, but if, after that stage, the bill fails to obtain the support of a minister in order to obtain a royal recommendation, that is it. The Speaker will then rule the bill out of order because it violates the Constitution and the Standing Orders.
So, I wonder why energy, time and money would be invested to examine the expenditures associated with the adoption of private members' bills, when the Speaker will ultimately decide.