So I believe I've "sort of" resolved my issue!
I can at least get the VMs to boot and see the Adaptec 7805 in ESXi, but I am still doing some preliminary benchmarkings in terms of performance (more to come on that).
The trick? Editing the .vmx file manually and adding the following line for the PCIe card being passed-through:
pciPassthru0.msiEnabled = "FALSE"