Re: Sims 3 won't launch after merging Origin accounts
@MarLove6 You're absolutely correct that an account merge can lead to an invalid base game CD key and the problem you're seeing. I'm not sure why the support rep said you needed to uninstall, unless he replaced your base game key. (In that case, you'd need to uninstall and reinstall to get the new key in your computer's registry.)
You can try going back to support; hopefully a different rep will recognize the issue. Please let me know how that goes. In the interim, I'll ask around if there's something else you'd need to say or do and let you know what I hear. It might take a day or two though, meaning going through support again is probably going to get faster results.