If you had one in the plane and had tested it out several times VFR and it worked well, would you consider it in an emergency like this?
If I had an emergency (complete electrical failure in IMC, something of that nature) and had no other options due to fuel or whatever, I would use everything at my disposal which would likely mean flying an approach with an un-approved GPS or my handheld VOR reciever.
On another note it would be a tough call for the pilot if he had to pick between options of a radar approach at a field he wasn't quite sure he could make, or shoot an ILS with a handheld at a nearby field. I'd probably take the ILS if I had tested it several times and knew how to use it. At least if he went missed he would have a few minutes to fly to a remote area and punch out
Couple thoughts....
1) For your example, if I lose both gens and both batteries in the Hornet, I am ejecting. You lose the flight control system with no electricity, so that is game over. It will depart controlled flight within a couple of seconds after that. One scenario where something like the handheld could be useful would be nordo in hard IMC to minimums. But if that happened and I had enough fuel to go somewhere else, I would go to there. I never walk to the jet without having a general idea of where to turn to if I can't land back home (or at my destination).....ie a suitable and legal weather divert. If I gooned away my fuel reserves enough to be shooting a PAR to mins without a divert option, then I have really failed at my job......but in that specific scenario, yes, I would try the handheld if I knew it worked from previous experience. Honestly though, before I did that, I would use the air to ground radar, designate the end of the runway, and fly a normal approach using that. Not an option everywhere (specifically due to terrain considerations), but around here, I can do that without any real danger to myself.
2) Like McFly said, the red tape and the operational testing costs prohibit cool things like this. Our version of ASRS (called ASAP) is always flooded with anecdotal stories of why a civilian ILS would have prevented some safety of flight related hazard. Every NATOPS (our procedures standardization program) conference brings the issue up as well. It is a high visibility complaint within the community to say the least. And yet, each year, it is rejected due to lack of funds, interest, or some combination of both.
3) If I am going to use an unauthorized gadget to get me out of a corner of poor airmanship, I had better get out of it without breaking a jet. If I planted a jet into a house and ejected just prior due to some weird interaction of a handheld ILS, I would be done flying. I know that isn't a reason to not use it in extremis, but that maybe sort of explains the mindset to trying things that aren't on the ok to use list. Plenty of other ways (that have saved many a Hornet in it's 30 yr career) to avoid needing to use one to save an aircraft. Good planning, good situational awareness in flight, and good decision making are all high on that list.
4) If I were presented with the PAR to mins vs ILS to not mins option, it would just depend on how good that alternate wx was and if I had the fuel to get there. I'd take the handheld ILS to maybe TACAN mins, but if some other field had that, I'd just fly the TACAN. Otherwise, if it were iffy both places, I'd take the PAR hands down. I routinely fly PAR's to touchdown in VMC, just to see how good they do. Generally they get me onto the runway in a safe manner. If it were IMC, I would be comfortable flying one to the runway (as in below mins) if it were my only option.