Can rman backup a subset of a table

If logging with RMAN, you'd have to set the filename yourself - there's no wildcard. But there is an APPEND option.

spool log to 'my_log_file' append

Similarly, invoking RMAN with the APPEND option causes it to append to any log file specified.

Our backup scripts do the log maintenance at the OS level:

export LOGPREFIX=$ORACLE_ADMIN/$ORACLE_SID/rman/log/my_backup_script
export MYLOG=${LOGPREFIX}.$$   # add the pid to the log file name
rman log=${MYLOG} append <<EOF
... various RMAN commands
EOF

#Perform error checking and notification
fgrep -i RMAN- ${MYLOG}
if [ $? -eq 0 ]
then # we had a failure - email me
   mail -s "$0 : failure for `date +%m/%d/%y`" me@myself_and_i.com < ${MYLOG}
fi
# add the incremental log to the permanent log
cat ${MYLOG} >> ${LOGPREFIX}.log
# clean up the incremental log
/bin/rm -f ${MYLOG}

It sounds like you need to set up a recovery catalog in a central database, which you can read up on how to set up a Recovery Catalog and maintain it from the Oracle Database 12c Docs.

Once you have the recovery catalog, then dropped database backup records are still maintained with their respective retention policies:

Connect to the recovery catalog:

sqlplus rman/ @catdb

Get the dbid for the database in question:

select dbid, name from rc_database order by name;

Point the recovery catalog to the database using the DBID from the previous query:

call dbms_rcvman.setdatabase(null,null,null,<DBID>,null);

List the backups for that database:

select * from rc_backup_files;