Hi,
Of course, this scenario is not exactly standard flow in FM, but you can apply it, if you want. However, you should be very careful. For example, I don't understand your point about not controlling the budget after PO. How you will close your FM commitments, then? I'd suggest you to create FM update on GR only and not on invoice. In this case, GR will close PO's commitment without impacting the budget. As for invoice, it will be posted on non-budgetary FM assignment, which you can control through FMDERIVE.
In any case, as general recommendation, these 'experiments' should be done where they belong, i.e. in Test environment.
Because, I have an impression that all these modifications are applied directly in Production, if I judge by your other post about PR. 
Regards,
Eli