Moved to "Arduino Nixie Clock Kit Support Forum" where it's on topic and will get more views.
To the best of my knowledge no stripped down version of the software exists. Closest thing would be the Test Mode that runs on a freshly flashed board and/or after doing a factory reset. You've probably encountered the test mode: all 6 tubes count from 0 - 9 repeatedly with LEDs cycling through primary colors, then shuts off after 1 minute.
The 10 hour tube not lighting sounds familiar. Exactly which boards do you have - Rev. 3 or Rev. 3.01? This is important as several I/O changes were made from Rev. 3 to 3.01. The revision should be printed on the boards, kind of in between the 10sec and 1min tubes. You need to check in the code and maybe change a #define that is right after all the includes. #define FWV2 for a 3.01 board or #define REV3 for a 3 board.
Not sure why you are having a hard time flashing. How are you going about it? By far the easiest way is to stick with the chips that came with the kits (if they run at all they're 99.9% certainly not bad chips) because the bootloader is already on them and you don't have to mess with fuses. Plug the chip into an UNO and flash away from the Arduino IDE.
If you insist on flashing "virgin" chips you need some sort of ISP that lets you talk to the Atmega's SPI interface. I use an old AVRISP Mk.II with good results.
1) Set fuses. From command line say:
avrdude -v -c avrispmkII -p m328p -P usb -B 4 -b 19200 -U lfuse:w:0xFF:m -U hfuse:w:0xDE:m -U efuse:w:0x05:m
but change the "-c" option to match your particular ISP.
2) Cycle power to the chip!!
3) In Arduino IDE make sure that Tools -> Programmer is set to match your ISP.
4) Do Tools -> Burn Boot Loader
5) Cycle power to the chip!!
6) Now flash like you would do any other Arduino.