fix(apigateway): Lambda integration for imported functions (#8870)

Use `instanceof` to identify real functions vs imported ones and appropriately use whats needed to extract the function name.

Fixes #8869

----

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

packages/@aws-cdk/aws-apigateway/lib/integrations/lambda.ts

@@ -73,10 +73,20 @@ export class LambdaIntegration extends AwsIntegration {
       });
     }
 
-    const cfnFunction = this.handler.node.defaultChild as lambda.CfnFunction;
+    let functionName;
+
+    if (this.handler instanceof lambda.Function) {
+      // if not imported, extract the name from the CFN layer to reach
+      // the literal value if it is given (rather than a token)
+      functionName = (this.handler.node.defaultChild as lambda.CfnFunction).functionName;
+    } else {
+      // imported, just take the function name.
+      functionName = this.handler.functionName;
+    }
+
     let deploymentToken;
-    if (!Token.isUnresolved(cfnFunction.functionName)) {
-      deploymentToken = JSON.stringify({ functionName: cfnFunction.functionName });
+    if (!Token.isUnresolved(functionName)) {
+      deploymentToken = JSON.stringify({ functionName });
     }
     return {
       ...bindResult,

packages/@aws-cdk/aws-apigateway/test/test.lambda.ts

@@ -259,4 +259,23 @@ export = {
 
     test.done();
   },
+
+  'bind works for integration with imported functions'(test: Test) {
+    // GIVEN
+    const stack = new cdk.Stack();
+    const restapi = new apigateway.RestApi(stack, 'RestApi');
+    const method = restapi.root.addMethod('ANY');
+    const handler = lambda.Function.fromFunctionArn(stack, 'MyFunc', 'arn:aws:lambda:region:account:function:myfunc');
+    const integration = new apigateway.LambdaIntegration(handler);
+
+    // WHEN
+    const bindResult = integration.bind(method);
+
+    // the deployment token should be defined since the function name
+    // should be a literal string.
+    test.equal(bindResult?.deploymentToken, JSON.stringify({functionName: 'myfunc'}));
+
+    test.done();
+  },
+
 };