I checked with support and there is no reasonable way to make this work currently using event handlers. They suggested a way the error could be completely disabled by deleting some files in the install directory, but that really doesn't resolve the issue since I only need to suppress it temporarily. My workaround ended up being to disable the Log itself when executing the small portion of code that would sometimes cause this error to appear when there didn't appear to be an actual crash, then re-enable the log after.
I logged request to make the OnLogError event handler work correctly in this case: https://community.smartbear.com/t5/TestComplete-Feature-Requests/OnLogError-event-handler-should-handle-Process-Crashed-message/idi-p/233461#M2600
