All your employer has to do is open a Colorado withholding account, which is free for them to do. See this:
Withholding Accounts | Department of Revenue - Taxation (colorado.gov)
If you can't persuade them, you definitely should make quarterly estimated tax payments to CO. Otherwise you'll be on the hook at year's end for a large tax bill, possibly including penalties. As @Hal_Al said, it's certainly the easier thing for you to do.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
