Update on my earlier post...
I thought my code would set the IRQRSTEN bit, but when I stepped through the debug code, I could see it was not. This is because that register only allows writing to it once during a change period.
So, by combining two lines into one, I was able to get my interrupts working.
WDOG_STCTRLH |= WDOG_STCTRLH_WDOGEN_MASK | WDOG_STCTRLH_IRQRSTEN_MASK;