Hi Matt,
Yes, this is possible to do. There are a couple of settings/configurations that need to be in place for custom statuses to sync between a request and its resolving object:
First, you need to have enabled the setting to sync statuses in your task and issue preferences:

Secondly, for the custom statuses that you have set up, you must ensure that the status key used for the status on each object is identical. In the below screenshot, the "Awaiting Feedback" status for issues has a key of "AWF". If I wanted tasks in an Awaiting Feedback status to sync to issues, I would need to ensure that the Awaiting Feedback status for tasks is also set to AWF

When both of these settings are applied, you should find that the status of the resolving object (task in your case) will sync to the resolvable object (issue).
You can read more about these settings and configurations here on Experience League.
Best Regards
Rich