The choice of sync service and other editors is not simple, I’m afraid.
It may be that iA Writer cannot use files in Backup and Sync, and instead you have to find a way to put your markdown files in Google Drive proper.
Other editors also work with different sync services differently. Best to review discussions like e.g., How do I work with Obsidian on Mobile?
And a search here for “Google Drive” brings up a variety of threads that may be helpful.