I was actually able top resolve this by doing the following:

In the main menu program (PRG), called SYSMENU, I put in some code that was as follows:

IF WEXIST("SYSMENU")
  Activate Window SYSMENU noshow
  @1, 21 SAY SYS_NAME COLOR SCHEME 7
ENDIF

After I changed Company Names from the COMPANY SCREEN and thus paths, when the company screen went away and I got back into that screen, the value of SYS_NAME was changed showing the company's appropriate name. This isn't perfect, but it works every time. The reason for the WEXIST() is that the COMPANY SCREEN comes before SYSMENU in its calling order.

What I did was find the SYSMENU window, temporarily activated it, then did an @ SAY again for the COMPANY NAME, which simulates a refresh. When I left the COMPANY screen, I was back on the SYSMENU screen where the COMPANY NAME appropriately displayed the correct company name based on my choice in the COMPANY screen.