I am sorry for your frustration. If you were not going to be on the title of the property, then I could see them denying your request, but not for this. I know that you are not buying the home outright, but what I meant by that was, if you were, then there wouldn't be a loan for you to be on or off, so would they still say no?
What you are looking for (and many persons do when they want to be absolutely sure of something) is a law that is written the way you want it written to affirm your position. Unfortunately it doesn't exist that way. You are left with a law (regulation) that says what it does but not specifically what you need it to say to prove that you are correct. What the 401 company is doing is hiding behind the IRS and making them the bad guy. Many companies do this for various reasons. (Currently, I am fighting with my mortgage company who is telling me that they are following Freddie Mac regulations but they are not, they are making up their own rules and blaming the outcome on another entity---so I continue to fight)
I recommend a certified letter stating your postition and quoting the Treasury regulation that I gave you. Challenge them to show you where in the IRS regulations it states that you have to be on a mortgage loan for 401k hardship distribution rules to apply. You may also wish to report them to BBB, or whatever regulatory body applies (whether they are a bank, investment company, etc.)
I'm sorry I don't have a better answer for you, but I would pursue this as hard as you can---because you are simply right!!