You could have cron run a script on every Monday and have the script check whether the day of the month is 1 to 7.

This check can either be integrated in your main script or you can write a wrapper script so that you do not have to make this change to the main script so that it can run.

 if [[ $(date +%d) =~ 0[1-7] ]]; then
 : run script
 fi

Checking both day of week and day of month:

 if [[ $(date +%w%d) =~ 10[1-7] ]]; then
 : run script
 fi