chore(lambda): resolve unable to reference AuthType from FunctionUrl (#31590)

### Reason for this change

I will resolve this issue: #31339 (comment). In the current FunctionURL implementation, there is no way to access AuthType, and therefore, when writing logic that depends on AuthType, there is no method to reference it.

### Description of changes

I will fix the construct to allow access to functionurl.AuthType

### Description of how you validated changes

adding unittest and integ-test re-run.

### Reason for Exemption:
The fix introduces no changes to the resources being created.

### Clarification Request:
This fix only makes the internal property authType of the L2 Construct publicly accessible, and does not introduce any differences in the created resources.

### Checklist
- [x] My code adheres to the [CONTRIBUTING GUIDE](https://github.com/aws/aws-cdk/blob/main/CONTRIBUTING.md) and [DESIGN GUIDELINES](https://github.com/aws/aws-cdk/blob/main/docs/DESIGN_GUIDELINES.md)

----

*By submitting this pull request, I confirm that my contribution is made under the terms of the Apache-2.0 license*

Author: watany-dev

packages/@aws-cdk-testing/framework-integ/test/aws-lambda/test/integ.function-url.js.snapshot/IntegTestDefaultTestDeployAssertE3E7D2A4.assets.json

@@ -37,13 +37,13 @@
     "Code": {
      "ZipFile": "def handler(event, context):\n  return \"success\""
     },
+    "Handler": "index.handler",
     "Role": {
      "Fn::GetAtt": [
       "IamAuthFunctionUrlsServiceRole35DF9DE0",
       "Arn"
      ]
     },
-    "Handler": "index.handler",
     "Runtime": "python3.10"
    },
    "DependsOn": [
@@ -99,13 +99,13 @@
     "Code": {
      "ZipFile": "def handler(event, context):\n  return \"success\""
     },
+    "Handler": "index.handler",
     "Role": {
      "Fn::GetAtt": [
       "NoAuthFunctionUrlsServiceRole7247E6F2",
       "Arn"
      ]
     },
-    "Handler": "index.handler",
     "Runtime": "python3.10"
    },
    "DependsOn": [
@@ -134,8 +134,8 @@
       "Arn"
      ]
     },
-    "Principal": "*",
-    "FunctionUrlAuthType": "NONE"
+    "FunctionUrlAuthType": "NONE",
+    "Principal": "*"
    }
   },
   "CorsFunctionUrlsServiceRole6227B597": {
@@ -175,13 +175,13 @@
     "Code": {
      "ZipFile": "def handler(event, context):\n  return \"success\""
     },
+    "Handler": "index.handler",
     "Role": {
      "Fn::GetAtt": [
       "CorsFunctionUrlsServiceRole6227B597",
       "Arn"
      ]
     },
-    "Handler": "index.handler",
     "Runtime": "python3.10"
    },
    "DependsOn": [
@@ -192,19 +192,19 @@
    "Type": "AWS::Lambda::Url",
    "Properties": {
     "AuthType": "NONE",
-    "TargetFunctionArn": {
-     "Fn::GetAtt": [
-      "CorsFunctionUrlsD81CF424",
-      "Arn"
-     ]
-    },
     "Cors": {
      "AllowMethods": [
       "*"
      ],
      "AllowOrigins": [
       "https://example.com"
      ]
+    },
+    "TargetFunctionArn": {
+     "Fn::GetAtt": [
+      "CorsFunctionUrlsD81CF424",
+      "Arn"
+     ]
     }
    }
   },
@@ -218,8 +218,8 @@
       "Arn"
      ]
     },
-    "Principal": "*",
-    "FunctionUrlAuthType": "NONE"
+    "FunctionUrlAuthType": "NONE",
+    "Principal": "*"
    }
   },
   "StreamFunctionUrlsServiceRoleAF76EC5D": {
@@ -259,13 +259,13 @@
     "Code": {
      "ZipFile": "def handler(event, context):\n  return \"success\""
     },
+    "Handler": "index.handler",
     "Role": {
      "Fn::GetAtt": [
       "StreamFunctionUrlsServiceRoleAF76EC5D",
       "Arn"
      ]
     },
-    "Handler": "index.handler",
     "Runtime": "python3.10"
    },
    "DependsOn": [
@@ -276,13 +276,13 @@
    "Type": "AWS::Lambda::Url",
    "Properties": {
     "AuthType": "NONE",
+    "InvokeMode": "RESPONSE_STREAM",
     "TargetFunctionArn": {
      "Fn::GetAtt": [
       "StreamFunctionUrlsAAB55C9C",
       "Arn"
      ]
-    },
-    "InvokeMode": "RESPONSE_STREAM"
+    }
    }
   },
   "StreamFunctionUrlsinvokefunctionurl4FD8689D": {
@@ -295,8 +295,8 @@
       "Arn"
      ]
     },
-    "Principal": "*",
-    "FunctionUrlAuthType": "NONE"
+    "FunctionUrlAuthType": "NONE",
+    "Principal": "*"
    }
   }
  },

packages/@aws-cdk-testing/framework-integ/test/aws-lambda/test/integ.function-url.js.snapshot/Stack1.assets.json

@@ -202,6 +202,11 @@ export class FunctionUrl extends Resource implements IFunctionUrl {
    */
   public readonly functionArn: string;
 
+  /**
+   * The authentication type used for this Function URL
+   */
+  public readonly authType: FunctionUrlAuthType;
+
   private readonly function: IFunction;
 
   constructor(scope: Construct, id: string, props: FunctionUrlProps) {
@@ -217,8 +222,10 @@ export class FunctionUrl extends Resource implements IFunctionUrl {
       ? { targetFunction: props.function.version.lambda, alias: props.function }
       : { targetFunction: props.function, alias: undefined };
 
+    this.authType = props.authType ?? FunctionUrlAuthType.AWS_IAM;
+
     const resource: CfnUrl = new CfnUrl(this, 'Resource', {
-      authType: props.authType ?? FunctionUrlAuthType.AWS_IAM,
+      authType: this.authType,
       cors: props.cors ? this.renderCors(props.cors) : undefined,
       invokeMode: props.invokeMode,
       targetFunctionArn: targetFunction.functionArn,

packages/@aws-cdk-testing/framework-integ/test/aws-lambda/test/integ.function-url.js.snapshot/Stack1.template.json

@@ -231,11 +231,13 @@ describe('FunctionUrl', () => {
     });
 
     // WHEN
-    fn.addFunctionUrl({
+    const fnUrl = fn.addFunctionUrl({
       authType: lambda.FunctionUrlAuthType.NONE,
       invokeMode: lambda.InvokeMode.BUFFERED,
     });
 
+    expect(fnUrl.authType).toBe(lambda.FunctionUrlAuthType.NONE);
+
     // THEN
     Template.fromStack(stack).hasResource('AWS::Lambda::Url', {
       Properties: {