当API Gateway代理到S3的PUT请求时遇到CORS问题，可以通过以下步骤解决：

1. 在API Gateway中创建一个资源和方法，并将方法设置为PUT请求。
2. 在API Gateway的方法设置中，启用CORS并配置CORS响应头。

{
  "swagger": "2.0",
  "info": {
    "title": "API Gateway to S3 CORS Example",
    "version": "1.0"
  },
  "paths": {
    "/s3proxy": {
      "put": {
        "responses": {
          "200": {
            "description": "PUT request successful"
          }
        },
        "x-amazon-apigateway-integration": {
          "type": "aws_proxy",
          "httpMethod": "PUT",
          "uri": "arn:aws:apigateway:us-east-1:s3:path/{bucket}/{key}",
          "credentials": "arn:aws:iam::123456789012:role/MyS3Role",
          "requestParameters": {
            "integration.request.path.bucket": "method.request.path.bucket",
            "integration.request.path.key": "method.request.path.key"
          },
          "responses": {
            "default": {
              "statusCode": "200"
            }
          }
        },
        "x-amazon-apigateway-cors": {
          "allowOrigins": ["*"],
          "allowMethods": ["PUT"],
          "allowHeaders": ["Content-Type"],
          "exposeHeaders": []
        },
        "parameters": [
          {
            "name": "bucket",
            "in": "path",
            "required": true,
            "type": "string"
          },
          {
            "name": "key",
            "in": "path",
            "required": true,
            "type": "string"
          }
        ]
      }
    }
  }
}

3. 在S3桶的CORS配置中，确保允许来自API Gateway的PUT请求的来源和HTTP方法。

  
    *
    PUT
    Content-Type
  

通过以上步骤，API Gateway代理到S3的PUT请求应该不再遇到CORS问题。