### 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_
connection_name = "This should fail with two errors 123467890123412341234123412341234"
        with pytest.raises(ClientError) as ctx:
            aws_client.events.create_connection(
                Name=connection_name,
                AuthorizationType="INVALID",
                AuthParameters={"BasicAuthParameters": {"Username": "user", "Password": "pass"}},
            ),
        assert ctx.value.response["ResponseMetadata"]["HTTPStatusCode"] == 400
        assert ctx.value.response["Error"]["Code"] == "ValidationException"
        message = ctx.value.response["Error"]["Message"]
        assert "3 validation errors" in message
        assert "must satisfy regular expression pattern" in message
        assert "must have length less than or equal to 64" in message
        assert "must satisfy enum value set: [BASIC, OAUTH_CLIENT_CREDENTIALS, API_KEY]" in message
    def test_put_events_with_target_firehose(self, aws_client):