write to SQL server from another server
Unfortunately using application pool identities between two different servers is not possible. App Pool identities are located on their host machines only - SQL server will be unaware of the application pool accounts - so you cannot add them explicitly to logon to the SQL server. They do however run as the Local Machine account over the network - in a domain environment you could give the application server's machine account permissions to the SQL server.

If you have a domain they could be joined to you could run the application pools under a domain identity; granting the domain account access to the SQL server.

This is my preferred method for a dev environment: Enable the "Local Account Token Filter Policy" using group policy. Cross create your accounts on both servers with same username/passwords. Run the application pools under the created accounts; you can grant the same account access on the SQL server; since the account exists on both servers you shouldn't have any issues.