How to check if more than an instance of file exists and remove it

Question

I have written a very simple script to make backup of my files and databases daily. This is my script:

mongodump --db DB
name=$( date +%F_%H_%M_%S )
tar zvcf /backup/$name-DB-db.tar.gz dump/
rm -rf dump
name=$( date +%F_%H_%M_%S )
tar zvcf /backup/$name-FILES-files.tar.gz /home/NAME/

It makes two tar.gz files each day, and after a week there will be 14 tar.gz files.

What I want is, suppose the below output of ls -lh:

2021-06-19_16_02_00-FILES-db.tar.gz
2021-06-19_17_02_00-FILES-db.tar.gz
2021-06-19_16_02_00-FILES-files.tar.gz
2021-06-19_17_02_00-FILES-files.tar.gz
2021-06-19_16_02_05-DB-db.tar.gz
2021-06-19_17_02_05-DB-db.tar.gz
2021-06-19_16_02_08-DB-files.tar.gz
2021-06-19_17_02_08-DB-files.tar.gz

What I want is:

Check if more than file {*FILES-db.tar.gz} exists, then remove the old ones and keep the last one. Else, skip.
Check if more than file {*DB-db.tar.gz} exists, then remove the old ones and keep the last one. Else, skip.

And the result of ls -lh after that should be something like this:

2021-06-19_17_02_00-FILES-db.tar.gz
2021-06-19_17_02_00-FILES-files.tar.gz
2021-06-19_17_02_05-DB-db.tar.gz
2021-06-19_17_02_08-DB-files.tar.gz

What function or commands should I use in my bash script?

I think it's better to use this function at the end of my bash script after making backup is done, but if there are better approaches and solutions, I'm eager to hear.

Update 1

/etc/logrotate.d/NAME-files:

/backup/*NAME-files.tar.gz {
        daily
        missingok
        rotate 1
        notifempty
}