### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

IaC = CDK.

Attempting to connect an API-GW endpoint directly to EventBridge.

```
      const awsIntegration = new apigateway.AwsIntegration({
      service: 'events',
      action: 'PutEvents',
      integrationHttpMethod: 'POST',
      options: {
        credentialsRole: role,
        requestTemplates: {
          'application/json': requestTemplate,
        },
        passthroughBehavior: apigateway.PassthroughBehavior.NEVER,
        integrationResponses: [
          successResponse,
          serverErrorResponse,
          clientErrorResponse,
        ],
      },
    });
```

Receiving response:

```
    {
      statusCode: 400,
      data: {
        Type: 'User',
        message: `Error invoking integration for API Gateway ID '1z7ky5zzcb': API Gateway AWS integration action URI "arn:aws:apigateway:ap-southeast-2:events:action/PutEvents", method "GET" not yet implemented`,
        __type: 'InvalidRequest'
      }
    }
```

Similar to bug reported:

* https://github.com/localstack/localstack/issues/5508

### Expected Behavior

* 200 response
* Event created in relevant Eventbus

### How are you starting LocalStack?

With a docker-compose file

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

docker run localstack/localstack

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

Calling the API endpoint using axios from jest tests.

### Environment

```markdown
- OS: MacOS
- LocalStack: latest
```


### Anything else?

_No response_
class LambdaIntegration(BackendIntegration):
    def _lambda_integration_uri(self, invocation_context: ApiInvocationContext):
        """
        https://docs.aws.amazon.com/apigateway/latest/developerguide/aws-api-gateway-stage-variables-reference.html
        """
        uri = (
            invocation_context.integration.get("uri")
            or invocation_context.integration.get("integrationUri")
            or ""
        )
        variables = {"stageVariables": invocation_context.stage_variables}
        uri = VtlTemplate().render_vtl(uri, variables)
        if ":lambda:path" in uri:
            uri = uri.split(":lambda:path")[1].split("functions/")[1].split("/invocations")[0]
        return uri
    def invoke(self, invocation_context: ApiInvocationContext):
        headers = helpers.create_invocation_headers(invocation_context)
        invocation_context.context = helpers.get_event_request_context(invocation_context)
        invocation_context.stage_variables = helpers.get_stage_variables(invocation_context)
        if invocation_context.authorizer_type:
            invocation_context.context["authorizer"] = invocation_context.authorizer_result
        func_arn = self._lambda_integration_uri(invocation_context)
        event = self.request_templates.render(invocation_context) or b""
        asynchronous = headers.get("X-Amz-Invocation-Type", "").strip("'") == "Event"
        try:
            result = call_lambda(
                function_arn=func_arn,
                event=to_bytes(event),
                asynchronous=asynchronous,
                invocation_context=invocation_context,