I think you're on the right track when it comes to using a timer. In fact, for your purposes, the old-fashioned REALLY simple mechanical timers will be your friends.

However, you're right, modern fans, especially tower fans, are a slave to fancy circuitry, and I honestly can't remember ever seeing one that doesn't have some kind of stupid electronic control rather than a simple on/off switch.

What you may need to do is void the heck out of your warranty, and just hard-wire the fan to stay on, bypassing all the controls.

Good luck!

EDIT: No sooner than I post this, than I go doing some looking, and find this, or even this somewhat nicer version, which might suit your purposes.