Probably. In a war I don't think they'd hesitate, since they've already demonstrated their willingness to cause huge clouds of LEO debris with simple tests. But the upside is that it's much less of a problem for the rest of humanity. Losing access to space for a few years is certainly not something we want, but in the context of a major war between two space powers it will be among the least of our worries.

Note also that "Kessler Syndrome" is not a binary condition. As debris gets worse, the expected lifespan of satellites in those orbits drops, and as does the success rate of launching rockets through the effected altitudes. However these don't drop off to zero immediately. If the satellite is important enough to justify the elevated risk and additional expense then you can simply plan to launch several of them to get one of them through. Manned missions would be a bad idea, but odds are good that vital weather satellites and the like would be kept aloft.

That depends on whether that military thinks they rely on space less than the other military. But likely there will always be at least one military that thinks it would be to their advantage.