I found the answer here:
WiX <util:CloseApplication> element not working
I made one tweak to the solution in the post above. I kill the application earlier in the install sequence so the window above doesn't appear.
<!-- Code to force termination of running program...MSIExec couldn't do it -->
<Property Id="QtExecCmdLine" Value='"[WindowsFolder]\System32\taskkill.exe" /F /IM "$(var.ProductName).exe"'/>
<CustomAction Id="APP.TaskClose" BinaryKey="WixCA" DllEntry="CAQuietExec" Execute="immediate" Return="ignore"/>
<InstallExecuteSequence>
<Custom Action="APP.TaskClose" After="LaunchConditions"/>
</InstallExecuteSequence>
If you are wondering what "$(var.ProductName).exe" is, I pass the exe name on the commandline because I'm creating several branded versions of the same program. Just substitute your exe name.
And yes, it is safe in this particular intance to do this. There is no data held in memory that could be lost.