I have some further information and I believe I have found the root cause of this issue.
During operation we are running the core off a 1.2V switching supply which we switch over to during boot. This switchover circuit is a make-before-break circuit with ~1ms of crossover time controlled by a single Vybrid GPIO which defaults to the Vybrid's internal LDO. When we issue a SW reset all of the GPIOs default to inputs causing the external pull on the core select IO to switch the core supply back to the Vybrid's LDO. This causes a small step in the core supply voltage as the two supplies are not exactly the same voltage. The Vybrid doesn't seem to like this during reset. I've tested switching during operation many times before (even though we don't plan on doing it) and we switch every boot and have never seen an issue. We've confirmed that If we switch the supply then issue the SW reset command I no longer see the multiple resets.
Note that we are seeing different values in the reset status register (SRC_SRSR) depending on what we do as well that was causing some other problems:
1. Power On Reset: 0x00FEFF65 (expected)
2. SW Reset without Switching supplies beforehand: 0x00001201 (unexpected)
3. SW Reset after Switching to the Vybrid's LDO: 0x00040000 (expected)
So I believe I have some new questions:
1. Is this behavior expected? Is it indicating a different design issue?
2. How long should I wait after switching supplies before issuing the SW reset command?
Below are some scope captures for further information:
Reset signal (green) and core supply (purple) when issuing a SW reset without switching the core supply first:

Reset signal (green) and core supply (purple) when issuing a SW reset after switching the core supply. Note that you can see the supply switchover towards the end of the capture:

Detailed captures of the core supply when switching between the LDO and switching supply. In this case the LDO seems to be at a slightly higher voltage (~50mV) than the switching supply:



