Yes, a support case can be expensive. But you have to weigh this against the alternatives. How much does it cost you to not being able to apply the CU? How much would it cost you to uninstall the instance and reinstall it and reapply service packs etc? Don't forget to weigh in the risk that you mess up more things so that you have to repair the other instances. Note, I'm not saying that this is necessarily more expensive than opening a case with Microsoft - if you are hobbyist working at home, maybe time is all you have. But if you are doing this as part of a full-time job, opening a case permit you spend time on all your other pressing things.
And, while maybe not that likely, this may be due to a bug in SQL Server or Windows. In such case, the support case will be without cost for you.
Before you open a case, you could try a few more last-ditch attempts: 1) Reboot windows. 2) Run SQL Server setup and try to run Repair.
Erland Sommarskog, SQL Server MVP, www.sommarskog.se