Power LEDs from wall outlet without transformer plug

... we would like to know if plugging [the LED strips] into an outlet (using a 12 VDC power supply) would be safe, given the speed/frequency that the LEDs will be turning on/off. Would it be a fire hazard?

To answer your question directly and simply: No.

Should you use AA's? No. They will die out very fast, given you could be drawing as much as maybe 1A current with both pedals and multiple LED strips.

The speed at which you turn the LEDs on and off is not a concern. The important thing is you wire everything well, and don't short anything out. Here's a quick list of things you should concern yourself with:

• The switching system on the bass pedals: Most pedals are almost completely metal, so when wiring things onto it, make sure not to have any bare wires touching the pedal. This is especially important, because somewhere on the pedal you're going to have to make a bare metal contact to act as a switch.
• The "Current Draw" of your power supply: Look at the "Current Draw" specs of whatever LED strips you choose. Make sure your power supply is rated for well above that, and account for each LED strip you use.

Something trivial to consider: If you play a mean double bass, chances are you may not see much of a "space" between the LEDs going on and off, they will just appear dimmer than if you were to hold the pedal down. This is more-or-less how LEDs like a Macbook's "snoozing" LED gets dimmer and brighter. They (big wave of the hands here, EE's look away) pulse the LED at different rates to change its brightness.

What you want is a strip that has clearly-delineated connections between segments such as those that the LED strips that Sparkfun sells has. This way you can be sure that the segments are individually usable.

Solid strip (Sparkfun product 12021):

Strip with individually-addressable LEDs (Sparkfun product 12027):