What is the fastest way to load PHP functions that are only used in one theme template?

This is a non issue if you use PHP 7.0+ that keeps all files compiled in memory (not 100% accurate but a good enough abstraction), and since all other PHP versions are not supported or are very near their End Of Life, you should upgrade to 7.0+ version if you are not on one.

Regardless, PHP execution is rarely the bottleneck from a performance POV, DB access is usually the bottleneck. It is more important to write your code in a way which is easier to understand and maintain than to look for saving one micro second in page generation.

If you do not have strong opinions of your own, just try to follow how core themes are structured. They do not necessarily have the best structure, but more people are familiar with them and it will be easier to get relevant help either as an advice, or as someone that help coding the theme.