I've seen the opposite in a bioreactor (about the most stable environment you can have). Variants slowly emerged and outcompeted each other until a variant emerged with a DNA repair pathway deleted. The spawns of that variant quickly out-competed the old ones and from then on there were ~10 dominant variants at any given time instead of the previous 1-3.
Because the environment was so stable, the genes did not need to "remember" previous states of the environment and were free to optimize for the current state. The way to do this is to increase the mutation rate, as you potentially delete old genes that are no longer necessary and create new genes that are right for the moment.
Edit: I suppose this points out a difference between "fast-changing" in a single direction and "fast-changing" that repeats states, such as seasonal changes.
Now consider that mutation is not a single "rate" across the genome. Instead, mutations are targeted to specific regions! Some genome sequences have mutation rates of 10^-3, compared to 10^-9 for most of the genome.
So, when considering mutation in relation to evolution, examine the genes with high mutation rates, especially in their regulatory regions. If fast-changing environmental FACTORS can influence evolution, then hypermutability may be advantageous in genes that interact with these factors. By targeting mutations to specific genomic regions, organisms can "bet-hedge" with specific traits, increasing the probability of survival during extreme environmental shifts.
Think about growth rates in relation to boom-bust food cycles, or neural development in relation to dynamic social conditions. What evolutionary strategies would be optimal for rapidly changing environmental factors? Increasing mutation rates genome-wide is not the answer, as you point out it leads to problems with viability. Thankfully, mutation rates are local and traits are bet-hedged against variable environments.
What a wonderfully variable species we are! We adapted to different environments worldwide! And it only took tens-of-thousands of years?! Amazing :)
I would also add that volume of reproduction and lifespan of an organism also plays a role. Even if your mutation rate is "low" but you reproduce a lot and have a short lifespan (think like fruit flies) then you could offset slow mutation because your generational mutations are happening quickly.
In short, mutation rate can't be the only factor that might be selected for given environments that might be experiencing huge amounts of change.
There is no reason to think that the rates important for adaptability/evolution have anything to do with those involving cancer and organ failure of non-gonadic tissue.
Which leads to an interesting point: higher mutation ratios lead to more diversity and faster evolution/adaptation, but also lead to a higher % of cells/embryos failing due to destructive mutations (such as cancer).
Similarly, lower mutation rates increase stability and decrease cell/embryo mortality rates, but at the cost of adaptability and evolution.
With that in mind, you'd expect that fast-changing environments would select for genes that increase mutation rates, leading to faster evolution but with higher mortality rates, and stable environments would select genes that decrease mutation rates, resulting in larger populations and lower mortality rates.
It is amazing to think that even the rate of mutations is probably an evolutionary response to the environment.