I know this is an older post, still wondering how it turned out...
Per VZ's comment, isn't a failed Cat state going to send the same signals as a missing Cat to the O2 sensor, affecting the ECM's (sic) readings/behavior, or more concisely, input/output?
My (limited) understanding is that the air pump adds oxygen to post combustion exhaust gas, that the Cat uses to burn off
un-burned hydrocarbons.
If your truck runs well without a Cat now, the O2 sensor may be sensing a dilution of the exhaust gas from the air pump, and the ECM is not affected.
I think the only way to find out is to run her without the air pump. You can actually remove the pulley and the turbine, and use a fan belt made for a Alt-Water Pump-Crank. UNLESS that is not the setup of a '87...
If she runs the same, what use is an air pump?