SQL Server FullText searching on text that is slightly obfuscated before insert
The full text catalog will only reflect the contents of a table's data.

Since it sounds like you're not using full text search yet, and you just want to start, you basically have 2 ways to do this:

Option 1: add columns with the de-obfuscated data, and use full text search on 
those. Say you have a column dbo.Customers.SavedText that is obfuscated - you add a dbo.Customers.SavedTextClear column with the original text. You can populate it and keep it up to date with triggers on insert and update. (Triggers usually have a bad reputation - rightfully so, you wanna minimize the work performed in them - but you asked for a solution that didn't require modifying the original queries inserting/updating into the table, and this is what it looks like. When you tie my hands, my artwork isn't quite as good, ha ha ho ho.)

Option 2: do full text search in something like Elasticsearch. Build an ETL process that exports the fields you want to search, de-obfuscates them, and puts them in a database that's tuned for full text search, like Elastic. You can read about how Stack Overflow approached this problem, although when you read it, I bet modifying your original queries is probably going to sound like a better solution. The only reason I mention it is that if you really need to scale out, this can be a less expensive approach long term.