+1 on Corbin March's answer, thats exactly what I was thinking. It sounds like your code-base is a bit ugly and it's going to take more than a single sprint cycle to clean it up.
So just like Corbin said,
- branch it into a "refactoring project"
- test your branch change
- promote to your test environment for QA testing
- incrementally merge it back into the trunk until the refactoring is done.
I'm sure you wouldn't have any problems selling this to your dev manager, if your PM's are having a hard time seeing it then explain to them that Rome wasn't built in a day and cleaning up all the garbage thrown into Rome's streets isn't going to be done in a day either. The refactoring will take a while but it'll be well worth it in the end in terms of easier maintenance, quicker enhancement releases, less production tickets, and a more fulfilled SLA.