I don't think you can use AppPool for remote SQL Server. Give machine account Domain\ComputerName$, access to SQL, or you can run AppPpool with domain user account (domain\username) and give this user access to SQL.
Domain\MachineName$ is more secure and maintainable as you do not have keep updating password for user every n months, and restricts to only one machine. If you have multiple IIS server then you have to go with domain\username method, and set AppPool to use this user account (Windows Auth).
I think You can also create group Domain group and add all (IIS) machine name on this group and give this group permission to SQL.