Too bad on the IP test but that helps narrow down the issue, and I'm glad to hear it's already opened with support (I assume those are the engineers you mentioned). That may be the best route considering, I'm not sure how easy resolving something like this remotely without accessing the system would be.
I am still curious about that business ISP, I've heard of things being blocked via the ISP itself before. Basically it would just surprise me a bit if the issue wasn't external to your environment. Could be an issue with the machine/client you're using to connect from work though but the intermittent part makes everything suspect.
Just to clarify, in all cases where it doesn't work you ARE able to log into the server via the client and it connects you to the desktop, you just then see a black screen right? If there are some situations where it doesn't connect you to the Security Server at all the distinction is important. The fact that you connect correctly after disabling, flushing dns on, and re-enabling the NIC is definitely strange, more so I feel like because the IP test didn't work and that you can connect from a different external network.
I'd focus on the network side of this equation given the evidence on hand, and I've had good experience with VMware support so hopefully they track it down for you. I can't obviously, but if I'm being honest I'd love to come work in that environment and check it out myself.
~Brandit