AWS - API Gateway - S3 integration, passing requestId to object as a tag

Question

I have API Gateway integrated with s3:

s3Integration = (props: CustomApiGatewayProps): AwsIntegration => {
    const accessRole = new Role(this, 'accessRole', {
      assumedBy: new ServicePrincipal('apigateway.amazonaws.com'),
      inlinePolicies: {
        {...policy to put object to s3 bucket...}
      },
    });

    return new AwsIntegration({
      service: 's3',
      integrationHttpMethod: 'PUT',
      path: `${props.customBucket.bucketName}/{key}`,
      options: {
        credentialsRole: accessRole,
        integrationResponses: [
          {
            statusCode: '200',
            responseTemplates: {
              'application/xml': `
              <ID>$context.requestId</ID>
            `,
            },
          },
        ],
        requestParameters: {
          'integration.request.path.key': 'method.request.path.key',
        },
      },
    });
  };

Calling api endpoint triggers putting object into s3 bucket. I'd like to send a response to the client with requestID taken from api gateway. This is a part which works. But I also want to add this requestID to the newly created s3 bucket as a tag. Do you have any idea how can I configure it? It is AWS CDK written in typescript. I tried this:

requestTemplates: {
   'application/json': JSON.stringify({
     Tagging: 'RequestId=$context.requestId',
   }),
},

but it overrides body sent by client.