CDK synth for only one stack errors when resources are missing for the other stack

My CDK app has two stacks defined, one for a prod environment and one for a dev environment. I have a CloudFront function defined in the stack with a different file path for dev and prod builds (the behavior of the function necessarily needs to be different for dev and prod, i.e. the dev function implements basic authentication so that the dev site will not be publicly accessible and I don't want that code on the prod side). So for example, the CloudFront function is defined as follows:

const cfViewerRequest = new cloudfront.Function(this, "MySite-CFViewerRequest", {
  code: cloudfront.FunctionCode.fromFile({filePath: `backend/functions/cf-viewer-request/build-${props.stage}/index.js`}),
  functionName: `MySite-CFViewerRequest-${props.stage}`,
  comment: `MySite-CFViewerRequest-${props.stage}`
});

The idea here is that when the "stage" prop is passed to the stack constructor (it is set to either "dev" or "prod"), the resulting file path for the function will either be "build-dev" or "build-prod". However, if I only build the dev version, I get an error ENOENT: no such file or directory, open 'backend/functions/cf-viewer-request/build-prod/index.js' when running cdk synth with the dev stack name, since only the dev resource was built. The same goes for other resources, such as the CloudFront deployment, which also has specific dev and prod build directories. How can I run CDK commands for the dev stack without having to build the prod resources beforehand?

Or is there a better way I should be going about this?

Here is the entry point code:

#!/usr/bin/env node
import "source-map-support/register";
import * as cdk from "aws-cdk-lib";
import { MySiteStack } from "../lib/my-site-stack";

const app = new cdk.App();
new MySiteStack(app, "MySiteDev", {
  env: { account: "012345678901", region: "us-east-1" },
  stage: "dev",
});

new MySiteStack(app, "MySiteProd", {
  env: { account: "109876543210", region: "us-east-1" },
  stage: "prod",
});