I just looked at your route. Every toll road you are traveling on accepts cash, with no posted plans to eliminate that soon. Your simplest option under the circumstances is to bring $20 in quarters and pay the tolls in cash. (When you pick up the car, you tell Alamo that you will be doing so, and if you want to be truly sure not to get double-charged, take a photo of the cash meter at each toll gate; the photo will be time-stamped, and if need be can be used as proof that you paid the tolls.)
PS: The cash roads do have manned toll booths that will make change from bills, but some of the exits are coin-basket only. You can pay with bills at the manned booths, but I just found it less hassle to hand my passenger the baggie of coins and have them count out and hand me what I needed. Toss the handful in the basket, wait for the green and roll on through.