INIT means that you want to override the existing backup file. If you are not using INIT, you will append to the existing file, so that file will keep growing. Given your use case, you should use INIT. Having multiple backups in the same file, is bound to cause confusion.
Normally, you write each backup to a distinct filename, but in your use case that may not be necessary. That depends on how your client wants to have it on their side.
As for the size of the transaction-log backup, I can understand if the first backup was big if you had not backed up the transaction log for a while. In that case, the transaction-log backup includes all changes since that previous log backup. That is, there is no relation to your most recent full backup.
The transaction-log backups are entirely discoupled from the full backups with one exception: When you take your first transaction log backup, it will go back to the very first full backup after you put the database in FULL recovery mode.
For your, quite special scenario, you would do this:
- As long as you are only sending full backups, set the database in simple recovery.
- Once you have completed all your MERGE statements so that you can make incremental updates, switch to full recovery.
- Take a full backup and send to customer.
- From now on, you send daily TRN log backups.
Note that if the client is busy and not apply the backup they get on the 22nd, they cannot just ignore that and apply the file they get on the 23rd. They must apply all transaction logs. And they must restore with the WITH STANDBY option.