ApiGatewayV1Api construct for REST API Gateway only supports Lambda integration type

[Bidzina]

I need to use HTTP_PROXY integration for REST api gateway method but ApiGatewayV1Api construct only supports LambdaIntegration.

export interface ApiGatewayV1ApiFunctionRouteProps<AuthorizerKeys = never> {
    function?: FunctionDefinition;
    authorizer?: "none" | "iam" | (string extends AuthorizerKeys ? Omit<AuthorizerKeys, "none" | "iam"> : AuthorizerKeys);
    authorizationScopes?: string[];
    cdk?: {
        method?: Omit<apig.MethodOptions, "authorizer" | "authorizationType" | "authorizationScopes">;
        integration?: apig.LambdaIntegrationOptions;
        /**
         * Use an existing Lambda function.
         */
        function?: lambda.IFunction;
    };
}