Hi Richard,
As far as I'm aware much like most CB's of old it shares a chassis with another radio being the Midland 98 Plus.
The manual can be found here :-
https://manual.cbmonitor.cz/cbradio/manualy/Midland/index.php
(Just select Midland 98 at the bottom of the list to download a PDF)
It doesn't look like there's an option to turn off the 'beep' but there is a pin marked as such on the circuit diagram at the end of the manual being pin 7 of the microprocessor.
Maybe this is what you're looking for as regards a 'surgical' fix and might be worth trying to ground it bringing it low or otherwise tie it high via a resistor to the processors power line. (There must be a reason to mark the line 'beep'?)
Getcha best spectacles out.
Hope that helps.
All the best,
Victor