No, there’s no legal proof required, obviously. The point is that “saving that part of PIP” isn’t what it’s supposed to be doing. Hence the name.
If you “need” to save it to get over the rounds, then there’s something not right with either spending habits or the daily living component of PIP.
That makes the entire arguing that you “sacrifice so much in benefits” argument asinine. You’re sacrificing the part that you’re “supposed to” be spending in bus and train tickets. Regardless of how much you want to argue that the “Mobility part” isn’t actually meant to help with that, despite it being written both on the DWP website and on the citizens advice webpage.
In fact, that entire part of PIP is tested. You only get it if you need help with getting around, so yes, it’s very much implied as to how to spend that money, because if you’re disabled but can get around, you don’t get it. If “mobility component” wasn’t obvious enough.