Using a bash test:
if [[ "$(LC_TIME=C date +%A)" == 'Thursday' && "$(date +%d)" -le 7 ]]; then
exit 1
fi
Note: Functionally this is no different than the test in your question.
Just add that to the top of your script. If it is the first thursday of the month the script will exit right away, otherwise it will run.
Alternatively you could put it right into your crontab entry, something like:
0 1 * * 1,4 [[ "$(LC_TIME=C date +\%A)" == 'Thursday' && "$(date +\%d)" -le 7 ]] || /path/to/script.sh