Have you looked at the Meade hand controller to see what the Autostar II is saying when this long beep is happening?
I know it 's likely VERY inconvenient, but that is an option.

The PIPE.log is coming from the ASCOM Pipe diagnostic tool that was mentioned earlier. It can record every interaction between ACP (or other programs) and the LX200 driver.
In the Pipe screen, you'll see something like this:

pipelog.png
The stuff in the lower window will also get written to a pipe log file, if it has permission to write in the directory it was run from.
That will have the stuff in it like:
RightAscension: 03:19:09
Declination: +90:00:00
Azimuth: 00:00:00
Altitude: 43:32:09
CanSetTracking: True
Tracking: False

And then maybe we can figure out what is going on.