I made up the minutes display section on a breadboard to make sure my plan would work. I used two '164s, two LED bar graph segments, and the '00 NAND gate. This is enough to prove the counting, daisychaining and resetting. I used my super old school arduino to provide power and a 1 pulse per second clock signal.

After correcting a few wiring errors it all worked! Happy with that. Although these aren't the LED segments I'll be using in the final version, it seemed like the brightness was fine using 500ohm resistors.