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 IP for this machine is 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.
- 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.
Install on-premises data gateway - Azure Logic Apps | Microsoft Docs
Access data sources on premises - Azure Logic Apps | Microsoft Docs
HTTP with Azure AD - Connectors | Microsoft Docs