Without knowing more about your F5's configuration I would suspect you're correct, something is misconfigured within the HTTP compression profile for the virtual server.
BIG-IP will take the accept-encoding header from the client, and inserts a content-encoding header for gzip or deflate determined by the original client request. If the virtual server is not setup for HTTP traffic or does not have an HTTP compression profile I could see you having issues with proxying the accept-encoding and returning the content-encoding header.
My BIG-IP v13 HTTP compression profile defaults to gzip for preferred compression method.
![enter image description here](../../images/3872787694.webp)
If your virtual sever has an HTTP profile and not an HTTP compression profile, start by copying the default to a new profile and toggling this. If you're BIG-IP is older than v11, this field is in the HTTP profile. Give this a whirl and see what happens.
Check out F5 KB15434 for more information on this and if all else fails, DevCentral Q&A may have some more details on this.