I would initially try to understand how the website actually tracks the fact that cookies are enabled or not. Likely it's just a try to set a pre-defined string to the browser cookie and then it tries to read it. There is still a slim chance that certain dynamic value is used instead of pre-defined string, but it's pretty unlikely IMO.

Here is an exact topic about it: How to detect that JavaScript and/or Cookies are disabled?

Which means that you can initially investigate which cookie name and value is used for it opening a website from a normal user browser.

So then set it manually on the synthetic test setup (exactly as @bwest showed).

It may probably help you to overcome this initial check.