You can remove the PCB if you wanted to, but it would require removing some soldered connections and making the same connections to the LEDs with wires. The LED assembly would have those connection wires sticking out of it making it just as wide as it was with the PCB on it. You would have to snip those shorter. Then, once you have soldered wires to those snipped connections, you would probably end up with a unit just as deep... but now you'll also have to find a place for a PCB in your case.
Long story short (I tend to ramble in my posts) it'd be easier just to deal with the voltmeter assembled. The unit is 9mm tall, while a AAA battery measures in at 10.5mm. So you can stick the entire unit inside a AAA box, without even cutting a hole in the box (I would advise cutting a hole in the box so you can see the readout without having to open the box). With a hole you'd gain maybe another 2 mm of room, leaving at least 3.5mm of space (and that's not even counting any extra space the box manufacturer has supplied for the
batteries to begin with).
Seeing that the meter has the leads on the side, you can easily solder wire to it.
The biggest problem I would foresee would be the width and the length of the LED. Anyway you put it in, you'll need to use at least two AAA battery slots to get it in.