Maybe someone will have a better method for achieving what you request, but you can do much of that with UserCmdCheckStatus=1 and DumpPreUserCmd, where the DumpPreUserCmd will check if it's time to start the backup, or any other arbitrary check, and the backup will only start if that check returns 0. Check the documentation for those parameters and revert if you need more help.
With regards to backing up every other weekend, perhaps you're better off setting BackupsDisable=1 and starting your backups with a cron job.