Just because it has all those problems now doesn’t mean it has to continue to have them. If you find yourself making a specific bug fix in the system that could benefit from, say, a new data layer, then create a new data layer. Just because the whole site doesn’t use it doesn’t mean you can’t start using one. Refactor as you need to during your bug fixes. And make sure you understand exactly what the code is doing before you change it.
Problem with code duplication? Pull it out into a class or utility library, in a central location next time you have to fix a bug in the duplicated code.
And, as already mentioned by other responders – start writing tests now. It may be hard if the code is a coupled as it sounds, but you can probably start somewhere.
There is no good reason to rewrite working code. However, if you are already fixing a bug, there is no reason you can’t rework that specific part of the code with a “better” design.