SQL Server - Automatically full backup for new databases which are auto-created with EntityFramework-Migrations
my suggestion is to replace your script with one that is using ola hallengren backup solution with the option ChangeBackupType='Y'
https://ola.hallengren.com/sql-server-backup.html

ChangeBackupType
Blockquote


Change the backup type if a differential or
transaction-log backup cannot be performed.
DatabaseBackup checks differential_base_lsn in
sys.master_files to determine whether a differential backup can be
performed. If a differential backup is not possible, then the database
is skipped by default. Alternatively, you can set ChangeBackupType to
Y to have a full backup performed instead.
DatabaseBackup checks last_log_backup_lsn in
sys.database_recovery_status to determine whether a transaction log
backup in full or bulk-logged recovery model can be performed. If a
transaction log backup is not possible, then the database is skipped
by default. Alternatively, you can set ChangeBackupType to Y to have a
differential or full backup performed instead.

Basically, if you have this set to Y, then when the scheduled tlog for all user_databases runs, then the script will take a full backup instead of just a tlog.