I agree with you that trying to change the minimums bug inside the FAF would be an unnecessary distraction, and I would not do so. In reality, after this thread, I think I will take
@RussR's recommendation and simply go missed in the highly unlikely event the approach degrades. I could simply change the altitude in my head that I'm calling out 500, 200, 100 to, but there's really no downside to simply going missed and setting up for the degraded approach as well as trying to figure out why the approach degraded, unless you've already had an electrical failure or an engine issue. The likelihood of those occurring at the same time is vanishingly small; but falling back to that manual altitude callout would still work.
The problem, of course, with the idea of "I'll just switch minimums if it happens" is that it's real life, and real life reactions and confusion happens.
I've lost GPS integrity several times, some of those which resulted in a downgrade from LPV to LNAV.
Now, this is often trained on in the simulator during recurrent, but there in the simulator you KNOW why it's happening - you don't even think about it. More likely, you'll say "oh shucks, there went that pesky LPV again, I guess I'll have to use LNAV minimums" and you continue. You probably don't even try to troubleshoot or consider that you did something wrong. No big deal.
But in the real airplane it is much more confusing. The last time it happened to me, fortunately we were VFR, but still both of us up front had several seconds of "okay, what did you do?" "Did I load the approach right?" "I wonder if someone in maintenance turned off SBAS for some reason" "maybe try to re-load it?", etc., on like that. It was definitely short-term disorienting, even VFR. Of course, since we were VFR we just continued visually, to learn later that it was a no-kidding solar storm affecting that area of the country, and we had taken off before the NOTAM went out.
Another time it happened to me, it turned out to be a corroded antenna connection. And that's the problem with the "just continue the approach" method - you don't really know WHY the GPS lost signal or whether it's going to get worse. Antenna connection? Ice on the antenna? Antenna fell off? Cabling issue? Software issue? Firmware update causing problems? Operator error? Even something like cutting in too close to the FAF can cause some boxes to not activate the CDI scaling correctly.
Continue the approach in IMC? Unless I know I'm about to break out anyway, no thank you. I'll take it up and troubleshoot in a nice lazy holding pattern somewhere.
The same is true of the "lose the glideslope, revert to localizer only" idea. Maybe there are a few fewer things that could have gone wrong, but still there's going to be a period of significant confusion. Especially with older CDIs, which just park the glideslope needle in the center if it loses signal - hey look I'm perfectly on glidepath all the way down! As a CFI I've seen pilots miss those "off" flags many times, it's not out of the question for them to assume the glideslope came back and they're right on path because of their amazing luck. (I think that's a type of confirmation bias, see the fatal Korean Air 801 crash at Guam for a great example of "seeing what you want to see".) And since like with the GPS you really don't know what's going on with the antennas/cables/radios/ground transmitters anyway, best to go missed and troubleshoot.
Again, pilots may think they know what they're going to do and how they're going to react when faced with a situation, but you really don't know until you're faced with it. And unfortunately in these scenarios, the normal training methods don't really help because you know you're in training and that stuff like that is going to happen.