Strip Resource from Request Path in AWS Lambda Proxy Integration - aws-lambda

Question

I have a lambda function and API gateway deployed through terraform. The API gateway has multiple resources under the top-level root resource / such as /first and /second. The resource for /first is set up with lambda-proxy integration (ANY and /first and ANY on /first/{proxy+}). The /second resource points to a different integration.
My lambda is an ASP.NET Core 3.1 with Lambda Custom Runtime generated initially with dotnet new serverless.AspNetCore30WebApi but later updated to 3.1 when support was added to AWS.
My issue is that when the lambda proxy integration transforms the request to be sent to the lambda function, it includes the full resource name (i.e. /first/Product instead of just /Product). Since my REST API in the lambda function only knows about Products and doesn't have a concept of what first is, I get 404 results for anything forwarded through the API gateway. I can't seem to find a place to strip the resource name from the path before it's forwarded on to the lambda function, because with the lambda proxy integration, it doesn't seem like you have a chance to do any request mapping templates or substitutions.
Thu Jun 03 12:24:52 UTC 2021 : HTTP Method: GET, Resource Path: /first/Product
Thu Jun 03 12:24:52 UTC 2021 : Method request path: {proxy=Product}
Thu Jun 03 12:24:52 UTC 2021 : Method request query string: {}
Thu Jun 03 12:24:52 UTC 2021 : Method request headers: {}
...
Thu Jun 03 12:24:52 UTC 2021 : Endpoint request body after transformations: <<relevant snippet parts formatted below>>
{
"resource": "/first/{proxy+}",
"path": "/first/Product",
"httpMethod": "GET",
"pathParameters": {
"proxy": "Product"
},
"requestContext": {
"resourcePath": "/first/{proxy+}",
"httpMethod": "GET",
"path": "/first/{proxy+}",
"protocol": "HTTP/1.1"
}
}