### 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_
NAME = re.compile(r"([a-zA-Z0-9_-]+)(.*)$", re.S)
@patch(airspeed.VariableExpression.parse, pass_target=False)
def parse_expr(self):
    self.part = self.next_element(ExtNameOrCall)
    with contextlib.suppress(airspeed.NoMatch):
        self.subexpression = self.next_element(airspeed.SubExpression)
class ReturnDirective(airspeed.EvaluateDirective):
    """Defines an airspeed VTL directive that supports `#return(...)` expressions"""
    START = re.compile(r"#return\b(.*)")
    def evaluate_raw(self, stream, namespace, loader):
        import json
        value = self.value.calculate(namespace, loader)
        str_value = str(value)
        
        if isinstance(value, dict):
            try:
                str_value = json.dumps(value)
            except Exception:
                pass
        stream.write(str_value)
@patch(airspeed.Block.parse, pass_target=False)
def parse(self):