When I started this project I did not have the tools I really needed to analyze exactly how I was driving the clock so I went with the thought that the initial clock used a 1.5v battery so I should use close to 1.5v. This led me to use a low voltage motor controller. This worked well for me and allowed me to complete v1.
I've since obtained an oscilloscope and decided to see if I could eliminate the motor controller and use the ATtiny85 IO ports directly or use a bipolar H-Bridge. I found that the direct use of the IO ports, with 100ohm resistors, worked mostly ok and used less power than the H-bridges. This dropped average current draw from 0.75ma to about 0.45ma.
Then I tested using PWM for the clock pulses. This dropped the current draw down to just under 0.25ma with an added side effect that fast-adjusting the clock, as in a time change, is faster and smoother. (30 times normal speed)