Mac boot menu not showing up with different keyboards
I also use a non-mac keyboard and have seen the issue where it skips the boot menu. The native keyboard works both from a cold boot and a soft reboot. The 3rd party keyboard only works from a cold boot and not from a reboot. You should be able to power down your Mac and power it back on and have the boot menu work with any keyboard.

I've also read a few places that it matters which USB port you use for the third party keyboard (for the boot menu specifically). You could try moving the keyboard around to see if you get the desired behavior. YMMV.