Your findings are exactly what I found when comparing the schematic and the pin out of the FET Q3. That the instructions to turn the ZVP3306A around are wrong and will connect the drain to +5v, causing the fets body diode to conduct draining the battery when on battery power.
You can see my post here on this issue along with instructions on how to reverse the fet so the source is connected to +5volts:viewtopic.php?f=41&t=43209
The 2N2907A (in a TO92 package P2N2907A) was the original transistor used. If I can find the source again I will post it for reference.
P2N2907A Datasheet:http://media.digikey.com/pdf/Data%20She ... 20Rev4.pdf
The ZVP2110A FET is a better fit for the clock at 230 mA and 8 ohm on resistance vs. 160 mA and 14 ohm on resistance for the ZVP3306A ) and it works good. I would suggest changing the fet in the parts to this one.
And of course the current fet datasheet for comparison.
Changing the 22 ohm resister to a 11 ohm resister or a straight jumper (easiest and won't hurt anything) in place of the 22 ohm resister and reportedly helps some clocks that have issues.
I think using a 22 ohm resister and the ZVP3306A (14 ohm on) is to much of a voltage drop on the bias for some tubes.
The current schematic on GitHub shows the FET Q3 as connected to the VCC of the MAXX vfd chip and to the 22 ohm resister. Q3 is switching +5 volt power on or off to the MAX vfd chip and also the tubes bias pin which is what the other end of R3 (the 22 ohm resister) is connected to. Gate drive for Q3 comes directly from pin 5 of the ATmega168. A low signal on the pin turns the fet on, and a high signal turns it off. I checked my recently ordered kit after I received it against the schematic and it looks to be the same so I don't think there have been any revisions to the board.
Couple other things I found
The fet is very static sensitive which will damage it causing it to not work correctly, so don't touch the leds with your fingers when handling it and it is best to not lay it on the work bench unless it is wood. I usually just drop it in an antistatic bag for safe keeping. A note should be put in the instructions about properly handling static sensitive devices such as fets, vfd chips, processors.
An Overcurrent will also damage it and I found damage in such a way as it will not fully turn on or off.
Excessive heat will damage the fet. It is much easier to damage it during desoldering than when soldering it. If the solder pads delaminate from the board during desoldering, too much heat was used for to long and the fet is likely bad. I recommend clipping the leds flush with the top of the board and desoldering the pieces left. There is usually enough led left to solder it back in the right way.