You want to call an external API from Logic Apps (Consumption) and your requirements states that you have to use a Static IP.
You don't want to use Integration Service Environment (ISE) nor APIM as a solution.
You have a Windows machine with a Static Outbound IP, or your Network Outbound IP is static.
You can use OPDGW (On-Premise Data Gateway) as a gateway between the Logic App and your API. The API will receive the request from the IP of the machine hosting the OPDGW. The IP of this machine should be static, or your Network outbound IPs for this machine are static.
The machine should have access to the external API, so before testing the final solution; you should test if you can call the API from that machine, then install the OPDGW on the machine, set up the OPDGW on the Azure Portal, then try to call the OPDGW from the Logic App using an HTTP with Azure AD Action.
In the Logic App Designer, add an HTTP With Azure AD Action with Invoke, and connect it to the OPDGW (On-premise data gateway). You can Anonymous login and use the base part of the API URL as the base URL. HTTP with Azure AD - Connectors | Microsoft Docs
After creating the connection; You can use GET as the method, or POST, or what your API requires, and enter the URL for the API.
Test your solution; You should be able to call your External API through the OPDGW and your API will receive the call from your machine/network static IP.