If 9mm piezo elements are too large, you're basically stuck with a magnetic sounder. If those are still not small enough, well, the cell phone and earbud industries have spent millions trying to make really small speakers and this is the best they've done. You need to try something different.
If the magnetic sounder requires too much power, you will simply need to provide it with less power. For every part that I've worked with (just one) and every datasheet I looked up (three), the required currents were for operation at rated voltages. I have a Star Micronics NFT-03A electromagnetic sounder on my EK-LM3S6965 board. That part has the same form factor (visually) as their other NFT parts, it's tiny at about 5mm by 5mm by 2mm, but there's no datasheet and it's not available through the ordinary distributors. Other parts like this CUI buzzer are available in similar packages. They usually require between 70 and 110 mA, but consider the sound pressure levels that they generate and the drive method. The CUI datasheet says:
current consumption | at rated voltage, 4,000 Hz square wave, ½ duty | 110 mA
Clearly, this will be less if you drive it at a lower voltage, it will likely be less if you drive it at a lower frequency, and it will be less if you drive it at a lower duty cycle. You said 'buzzer', so you are probably able to tolerate the different sound produced by driving it with a 1000 Hz rectangular wave at 10% duty cycle. If it can be quieter, then drive it at lower voltage (or apply an ultrasonic PWM to the driver).
As an alternative, consider beeping the sounder intermitently rather than providing a continuous tone. Even at 1 Hz beeps, the thermal overload protection in your power supply shouldn't kick in during brief applications of power. Remember, the easiest way to lower power consumption is to turn the thing off.