First off it "works" by a very loose definition of "works" then you will do nothing but dig a hole insisting to rewrite it. In the end it will be politically futile, business people don't care about the same things you think you care about now, they never will.
The opinion/argument that "it will eventually blow up in your face" is a "slippery slope fallacy", it may also never be a problem of significant monetary value to the company. So the "safe" thing to do is not do anything until there is actually a live production problem, then do as little as necessary to limit the impact and risk.
The time for quality control concerns was passed as soon as this code went into production, that is a point of no return.
Arguing to "fix" it by "rewriting" it will just cause you way more pain and suffering in the end, even if you "win" and get your way, every little delay or mistake on the businesses part will be blamed on you and your insistent rewrite, regardless of what is actually at fault. Your reputation will suffer and you will lose the war if not the battle.
"The Grand ReWrite" never works, never has, never will, even when sponsored by the business. CYA documentation and other political stuff like that will backfire completely in most cases; for the most ridiculous illogical reasons imaginable.
For example from about 15 years ago when I was contracting; I have months of email CYA threads to a manager about a project that if done the way he wanted it done was a guaranteed failure waiting to happen, preformance was never going to meet the requirements, after like the 10th meeting I was told "just do what I want I am the boss", well I did, when I completed the project as requested in the manner requested and it didn't meet the performance requirements by 2 orders of magnitude, I brought up my CYA "I told you so 10+ times" emails, I was then blamed in a big meeting for "not doing a good enough job convincing the inept management that they were wrong, when I brought up the do as I say, that was even worse evidence that I didn't do enough to convince them, it was decided that it was my fault the customer was unhappy".
If you have to CYA yourself have already lost! You have to have CYA sponsorship and support from upper management for CYA to work as an approach.
Second, experience will tell you that everything, but especially software can be solved with an additional layer of abstraction. Wrap what you have to touch so you lessen your impact and risk to breaking it worse, move on.
30 years of experience tells me you should suck it up, keep your head down and "complete" the project as quickly as possible and train your replacement to move on as quickly as possible and be the "hero" by giving them what they want and make it someone elses problem as soon as possible.