S3 to Glacier You can target S3 Glacier and S3 Glacier Deep Archive directly if you have the option to show storage classes enabled in Settings - Global Agent Options.
You can also set up the Lifecycle Rules at the storage level in the Managed Backup management console in Storage Management for the S3 storage account in question. You should only have to do this once and the data should transition. You can transfer to S3 Glacier after 1 day if you want - or leave the data in S3 - Standard for a little longer to make restores easier.
S3 Glacier is a little different than classic Glacier. We use S3 Glacier which has some benefits over the older design.
In any case, what you are proposing is not something I would recommend. Glacier is for Archival storage. If you're using it just to lower costs, I fear you will not be happy with performance and cost should you need a restore. I would think you are better off moving to a lower cost cloud storage provider instead since they will give you hot storage at a lower cost.
Or, if you stay with Amazon, there are ways to lower storage costs without moving directly to Glacier. You can use something like S3 Infrequent Access or S3 One Zone Infrequent Access as the target for backups. They require data stay for a minimum of 30 days and then you can use a lifecycle transition policy to move the data to Glacier after 30 Days (or longer if desired). But only move the data to Glacier if you are certain you are not going to need to restore except in an unusual case or emergency. Restoring from Glacier can be slow and expensive. It's not hot storage and should not be thought of that way. You should do some tests to understand speed and have a look at the AWS Calculator to understand costs.
https://calculator.aws/#/createCalculator/S3
S3- IA is about half the cost of S3 Standard. One Zone IA is a little less than that. Using services like Backblaze B2 would cost about $5 / TB. Wasabi would cost $6 / TB.