I ended up using the cdk-assume-role-credential-plugin to perform the task. The description of that plugin states:

This plugin allows the CDK CLI to automatically obtain AWS credentials from a stack's target AWS account. This means that you can run a single command (i.e. cdk synth) with a set of AWS credentials, and the CLI will determine the target AWS account for each stack and automatically obtain temporary credentials for the target AWS account by assuming a role in the account.

I wrote up a detailed tutorial on how to use this plugin to perform AWS cross-account deployments using CDK here: https://johntipper.org/aws-cdk-cross-account-deployments-with-cdk-pipelines-and-cdk-assume-role-credential-plugin/