An ATmega328 has three timers, one of which, Timer1, is a 16 bit timer. The Arduino Uno and Nano both use an ATmega328.

As you have a A4988 module, then an Arduino Uno with a CNC driver shield would be a simple option:

A smaller option would be an Arduino Nano with a CNC board:

An ATmega328 has three timers, one of which, Timer1, is a 16 bit timer. The Arduino Uno and Nano both use an ATmega328.

As you say that you have a A4988 module,

then an Arduino Uno with a CNC driver shield would be a simple option:

A smaller option would be an Arduino Nano with a CNC board:

You could power it via a LiPo, using a TP4056 module, which has both Battery (B) and Load (OUT) outputs - there are two types of modules out there, one of which does not not a load output. This is probably what you need

Your solar panel would connect to the + and - inputs.

Obviously, you would use as many, or few, TP4056 modules as needed, depending on the voltage requirement of the µController. A NodeMCU µController would only need one LiPo and one TP4056, whereas an Arduino Nano/Uno would need two of each. There are the Arduino Pro Micro and Pro Mini that only require 3 V which run at slower clock speeds, which would match your efficiency requirement.

For use with a solar panel, see How to autoregulate a TP4056 for maximum solar power extraction for ideas about that.