Should I mention this in my graduate applications?
It is good that you are showing the initiative to seek advice on your application, so please take my criticism of your proposal in the positive spirit in which it is intended. As a general rule, it is not likely to look good for you if you are already volunteering contextual reasons for underperformance (a.k.a. making excuses) before you have even set foot in the program. Though their teaching duties, academics have experience with certain students who consistently underperform, and offer excuses in lieu of improvement. As a result, this type of comment excusing your low performance on that element of the test is likely to be seen as a "red flag".
There are some instances where universities ask people to put lower performance into context, such as when they ask about successes "relative to opportunity". In this context you are invited to raise disadvantages that have constrained your performance, though they are still expected to be substantial issues, not trivial ones. It is usually not a good idea to raise contextual issues unless prompted to do so, and it can give the impression that you are more interested in explaining-away bad results than doing what is needed to get good results.
... I looked into getting approved to use my alternate layout, but it seemed like a good amount of time and paperwork would be involved, so I didn't go through with it.
Unfortunately this makes it a lot worse. The fact that it is possible to put in some administrative effort to get the resources you needed, and you chose not to do this, is more likely to raise a negative inference than a positive one. If there were follow-up questions on this, and it came to light that you eschewed administrative work that was needed to get resources you needed to do good work, that would be seen as a serious negative, and could sink your application. PhD candidates are expected to be able to jump through administrative hoops to progress their work, and anything that indicates resistance to this in the application stage is going to hurt you. Even without that worst-case scenario, mentioning the issue is likely to focus attention on your low score, rather than focussing attention on the strengths of your application.
If you are not happy with your GRE score, you should consider re-sitting the test. If you need a particular piece of equipment to do this effectively, then do the administrative work required to get that. Good luck.