Yes, unfortunately I knowbthat Linux drivers are not compatible. But I have also read that Broadcom network card should be supported by ESXI.
I was hoping that maybe someone here found some sort of workaround for this issue...
Eventually if nobody has any suggestion for this issue, what would your approach be? Purchasing a PCI network card? Ofcourse after making surw it ia on the HCL 
. Or there are known issues with PCI cards? I am not sure, but I think I remember reading a topic with a fellow who ran into some issues with them.
The question is still open, maybe someone will suggest something still, without having to buy a new card...