User-declared exceptions can only be locally handled if they are locally declared. If they are declared in another package (even a local package), any raise will call the Last Chance handler and a local handler (if present) will be ignored.
This is believed to be related to
Standard exceptions such as Program_Error do not have this limitation : they may be locally handled, or handled by the Last Chance handler.
In either case, a local handler may not re-raise, as this is prohibited by pragma No_Exception_Propagation.
Use standard exceptions, or
Declare exceptions locally (in the subprogram which will raise and handle them),or
Make the exceptions locally visible (via a Use clause on their package), or
Don't handle exceptions locally.
Fix may not be possible without a better resolution to the upstream bug.