I'm pretty sure any judge is going to give them broad leeway to the business to self-determine the manner in which they secure their electronic systems.
Where your argument disintegrates (or coalesces) is what may be blindsiding you: There's more than one way to pay a bill. Or there ought to be.
The court system is an entity that runs on paper. And it runs on physical service. The court is going to have zero sympathy for the argument (from the college or a millennial) that "online is the only way to pay a bill". That will be seen as an idle preference and certainly not court enforceable.
I mention this because this is the perspective from which the Court will see your problem:
For over a century, the normal way of doing billing is for a business to mail a paper bill (typically once a month on specific dates) and grant 20-30 days for the recipient to pay it by postal mail. The time is for postal transit both ways, and to allow the customer to gather his mail and sit down and pay bills at sane intervals (e.g. twice a month). If you look closely at e-bill mechanisms, you'll see they are abstractions of this.
As such, taking a 15 day sabbatical isn't a problem - just check your mail and pay every bill you have, then do it again promptly on return. You can extend this to 45 days if you know what bills you are expecting. Not knowing which bills to expect is a bit alarming!
If you overpay a service bill such as a gas or insurance bill, the money is still yours. It is carried as a credit on your account, and applied to future bills. The college should do exactly the same thing. (I only pay my gas bill about once a year.)
Also, once you become a student and enter normal billing, it's likely the bill comes due after you've started the service. That makes it a debt. Cash is legal tender for all debts public and private which means the university cannot refuse.
So -- the questions that will come up in court:
- Why can't you act within the normal billing cycles (as I describe above)?
- Why can't you pre-pay expected bills (again as I describe above)?
- Why are you frequently being caught by surprise by bills (are you not aware when you contract services)? This makes you look oblivious.
- Why can't you receive bills by paper mail and pay by check? (no 2-factor authentication).
- Why can't you walk into the billing office, ask for a printout of your bill, and drop off or mail a check or cash?
I don't think you'll have credible answers for those questions. If you do, I'd say you have a case.