Can't pull down on +in, or up on -in, for the same result? It's rail-to-rail output, forcing the output to a rail does the same thing.
If the inputs need to be essentially isolated (no injected current allowed), you might add another stage as a buffer, then a series resistor and a pull-up/down there. Instrumentation amplifier configuration for example.
Can also look into gated type amps, though I don't know offhand if anyone makes one in zero-drift, or with nearly as high voltage (how much voltage do you really need?..).
Shorting should be okay as it's limited and thermally protected, but again, it's not a good idea to rely on. They don't give min/max current limits, perhaps because they're all over the place. So you might not be able to design in adequate heatsinking, to avoid the thermal limit, for example.
LDO regulators do this sort of thing with their current limits; it's a ballpark range, a safety feature, not an operating feature. It just means it won't explode instantly and you have some time to deal with the fault, but that time might be a hundred microseconds, it might be seconds. Obviously, this amp is delivering a lot less power than an LDO, and as long as the thermal protection does its job, it shouldn't release the magic smoke, but thermal cycling isn't great for recovery time or longevity.
Can also consider a series analog switch, instead of brutishly shunting it.
Tim