### 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_
return self.value
    def params(self, name=None):
        if not name:
            return self.parameters
        for k in ["path", "querystring", "header"]:
            if val := self.parameters.get(k).get(name):
                return val
        return ""
    def __getattr__(self, name):
        return self.value.get(name)
    def __repr__(self):
        return "$input"
class ApiGatewayVtlTemplate(VtlTemplate):
    """Util class for rendering VTL templates with API Gateway specific extensions"""
    def prepare_namespace(self, variables) -> Dict[str, Any]:
        namespace = super().prepare_namespace(variables)
        if stage_var := variables.get("stage_variables") or {}:
            namespace["stageVariables"] = stage_var
        input_var = variables.get("input") or {}
        variables = {
            "input": VelocityInput(input_var.get("body"), input_var.get("params")),
            "util": VelocityUtilApiGateway(),
        }
        namespace.update(variables)
        return namespace
class Templates: