There is a lot of useful information in the ICAO Flight Planning and Fuel Management Doc 9976 and there are also draft EASA docs on the net which attempt to mirror and expand on what ICAO are saying. Airline policies tend to be based on these documents but I don't think the training world has caught up with the 100+ pages of tech jargon. My basic understanding is that once airborne, the Commander's responsibility is to use all available information to manage fuel/energy to make a safe landing at a suitable aerodrome with Final Reserve Fuel remaining. The word safe seems to imply that there could be a less safe option and this would by default become applicable once the MAYDAY fuel call has been made or if committed to a particular airfield and stuck with having to land below minima. An unintended diversion to a military airfield is also classed as less than safe but it is in the controllers toolbox once it looks like Mayday fuels calls are a possibility. Why not consider Waddington for BHX or EMA say, if it is open and if there are problems and several Min fuel calls on freq?
There doesn't seem to be any reference to requiring a higher minima approach as per the planning rules so diverting to cat 2/3 is probably OK but perhaps not wise. Personally, I would go with the planning minima because as mentioned, many the other things that can go wrong which would preclude a certain approach are not likely to become apparent until on approach (flaps stuck?).
These documents do refer to the fact that it is far easier these days for pilots to gather information from other sources other than ATC so it should be possible to involve Ops with decision making but having looked at these docs, I can see how you got several different answers and why one ops controller might offer completely different advice than another.
In the scenario you mention, you have effectively committed and most ops manuals tend to describe the scenario of committing and lay down criteria, even if they don't actually call it that. I think most ops manuals require one lower level of minima (cat 1 for expected cat 2/3) for "committing" so I would probably use that in my justification.