Well, I finally got vSphere client to work on Crossover Version 12.2.1 (12.2.1.26977)!
Here is what I did:
1) Installed .net 3.5 sp1
2) Installed Internet Explorer 7 (probably not necessary, was trying something else)
3) Installed vSphere Client (VMware-viclient-all-5.1.0-1064113)
The last step exhibited the hcmon problem. I ran the installer again, which removed vSphere Client, then ran it one more time. This time installer complained again about hcmon but finished installation. vSphere client appeared as a valid application in Crossover and I could connect to my ESXI box.
Console, however, does not seem to work. But that does not bother me much.
Edit: Indeed, installing Internet Explorer was necessary. I removed it and vSphere Client would not render the GUI properly. Put the Explorer back in and it worked again!