This is because if you use the entire SSD as a partition to back up your Mac using Time Machine, Time Machine will automatically restrict writing to that partition to prevent the backup files from being tampered with.
How to fix:
You should not (and cannot) store files in the Time Machine volume. Instead, open Disk Utility and choose Add Partition/Volume on the disk. This allows you to have a place to store files on the disk.
Please follow the steps below to partition your SDD and use one of the partitions as TimeMachine so that the other partition can write files normally.
Please note that partitioning the SSD will clear all data in the SSD, so please back it up in advance!