I see that your requirement is to run the third party exe from your c# worker process which will be deployed to the Azure Container App. I am sharing a few suggestions here if that helps.
Firstly, to deploy your C# application worker process in an Azure container app, you can create a Dockerfile and then build the container image. Note that you can include your 3rd party exe path in docker file and use the COPY parameter to copy this within the container image while building this container image.
Then you can push this docker image to either container registry or dockerhub.
To run this 3rd party exe from your worker C# app within the container app, you can use the Process class in C# to start a new process and execute the exe file. See here and here.
This is tutorial to Build and deploy your app to Azure Container Apps see here.