How to add bucket policy to allow http API gateway routes to access S3 bucket data using S3 URL

Hi, I am experimenting the possibility to authenticate S3 file download using S3 URL, right now, I used HTTP Gateway that has a HTTP Integration that GET the S3 Object using S3 URL. Currently the bucket blocks all public access, I would be wondering if that's possible to write a bucket policy to allow API Gateway to access the object using the object URL from that integration. I tried this policy statement, but it seems it doesn't work, and I still got access denied

{
            "Sid": "AllowAccessFromAPIGateway",
            "Effect": "Allow",
            "Principal": {
                "Service": "apigateway.amazonaws.com"
            },
            "Action": [
                "s3:GetObject",
                "s3:PutObject"
            ],
            "Resource": "arn:aws:s3:::my-bucket/*",
            "Condition": {
                "ArnLike": {
                    "aws:SourceArn": "arn:aws:execute-api:us-east-1:account-id:api-id/$default/GET/image"
                }
            }
        }