I may be misreading your question but in the reference manual for your mixer, the voltage listed is 110-240 Vac @50/60Hz. So I am not sure you really need 12Vac or where you get that from.
Anyway, let's say you still need 12 Vac. Since you've stated you don't have a lot experience with electronics, I think the best way to go about this, is getting as close as standalone solution as possible.
For example, you said you tried feeding 12Vdc to the mixer and it didn't work. As mentioned in the comments, it's possible that 12Vdc is not enough since the actual peak voltage of the sine wave is about 17-18 volts. Plus, you don't really have 12Vdc anymore after the rectifier stage—there will be 2 diode voltage drops (for a full-wave recitifier) so what you may actually be feeding to the dc regulators after that more like 10V-11V range.
One thing to try could be a boost converter after the 12Vdc so that you get 17Vdc-18Vdc out that (may need to be higher to account for the diode voltage drops mentioned before).
This is boost converter that may work is the MAX618, click here. It can take a voltage anywhere from 3V to 28V and can output up to 28V (adjustable with resistors). It also meets the current requirements (<1A)—there are some tables in the datasheet that show the typical output current you can have for an input voltage (for 12V in and 18V out, can have a load pulling ~850mA).
You said that when you tried feeding DC, you had 'weird' results, maybe it had to do with the lower voltage (12Vdc). I just thought I give an idea.
Trying to come up with circuit to convert 12Vdc to 12Vac is a lot more complex, in my opinion, and you said you have basic electronics knowledge.