Can threat to survival increase mutation rates in germline cells?
Well... since I can't delete this accepted answer... it's going to be a reverse ferret, to some extent. One 2014 review by Ram and Hadany lists a fair number of SIM occurrences outside of bacteria:

Stress-induced mutagenesis (SIM)—the increase of mutation rates in stressed or maladapted individuals—has been demonstrated in several species, including both prokaryotes and eukaryotes. [...] SIM has also been observed in yeast, algae, nematodes, flies and human cancer cells.

The one for algae (Goho and Bell, 2000) appears/claims to have been the fist:

Cultures of Chlamydomonas were exposed to a range of relatively mild stresses for a period of 24 h. These stresses comprised high and low temperatures, osmotic stress, low pH, starvation and toxic stress. Fitness was then assayed as the rate of division of isolated cells on agar. We found that there was a strong tendency for stressed cultures to have lower mean fitness and greater standardized variance in fitness than the negative controls which had been cultured throughout in unmodified minimal medium. The same tendency was shown, as expected, by positive controls which received mutagenic doses of ultraviolet irradiation. We concluded that the most reasonable interpretation of these observations is that mild stress increases the genomic rate of mutation.  This appears to be the first time that this phenomenon has been noticed in eukaryotes.

The paper cited for Drosophila, Sharp and Agrawal (2012)

Our results show that mutation rates are sensitive to genetic stress, such that individuals with low-quality genotypes will produce offspring of even lower genetic quality, in a mutational positive feedback loop. This type of variation in mutation rate is expected to alter a variety of predictions based on mutation load theory and accelerate adaptation to new environments. Positive mutational feedback could affect human health by increasing the rate of germline mutation, and possibly somatic mutation, in individuals of poor health because of genetic or environmental stress.

Clearly this one is quite bold in extrapolating its findings.
And long story short, the paper on nematodes (Matsuba et al., 2012) finds a temperature dependant mutation rate.
What seems to be the weak point in these paper, and perhaps why a more critical review of Lynch et al., 2016 of SIM doesn't mention them, is that no explicit mediating mechanism appear to have been identified in these studies on eukaryotes. In bacteria (e.g. the paper linked by the OP) how SIM works inside the cell is pretty well understood, there are in fact several mechanisms that all respond (convergently) to various forms of stress.
There's an acknowledgement (in the Drosophila) paper, that such mechanisms in eukaryotes might differ from bacteria

The sources and mechanisms underlying this variation have been best studied in microbes, but the sources of variation in microbes may differ from those in multicellular eukaryotes for several reasons. [...]
In animals, mutation rate varies among genotypes, although the functional sources of this variation are unknown. [...]

It does go on into some theories how it might work.
As for yeast, Rodriguez et al. (2012)

Mismatch repair (MMR) is a major DNA repair pathway in cells from all branches of life that removes replication errors in a strand-specific manner, such that mismatched nucleotides are preferentially removed from the newly replicated strand of DNA. Here we demonstrate a role for MMR in helping create new phenotypes in nondividing cells. We show that mispairs in yeast that escape MMR during replication can later be subject to MMR activity in a replication strand-independent manner in nondividing cells, resulting in either fully wild-type or mutant DNA sequence. In one case, this activity is responsible for what appears to be adaptive mutation. This replication strand-independent MMR activity could contribute to the formation of tumors arising in nondividing cells and could also contribute to mutagenesis observed during somatic hypermutation of Ig genes.

I guess a weak point of this paper vis-a-vis of SIM (which they only mention in passing) is that it's not obvious in their setup what the stress was. Basically they saw (adaptive) mutation-rate change (in response to the environment), but they don't pinpoint what exactly they think the stressor was. So this paper is a way the reverse of the other three I, i.e. the mechanism is clear, but stress is not.