OpenID Connect as Client and Resource server

Maybe you need to focus on why the client does not send the cookie ? Is it a normal web user with a browser in normal not incognito mode ?

The issue you describe sounds not apm one as it is described in the clientles mode article that you can try using:

https://my.f5.com/manage/s/article/K000137617 

Also if this is API traffic see 

https://techdocs.f5.com/en-us/bigip-16-1-0/big-ip-access-policy-manager-api-protection.html as the API protection mode is newer way than clientles mode.

Other than that 21.1 ads DCR but maybe in your case it will not help as this is dynamic client registration:

https://techdocs.f5.com/en-us/bigip-21-1-0/big-ip-access-policy-manager-oauth-configuration/using-apm-as-an-oauth-2-server.html