Nixie Bathroom Scales - Kit

I see this as more of a mechanical problem than an electronic problem. Why? Consider:

Displaying numbers generated in a microcontroller on Nixie tubes is "settled science" (lol). Look at almost any Nixie clock circuit: the hardware is only concerned with getting numbers on to tubes - time keeping, and all the rest, happens in software, A scale circuit would look quite similar to a clock circuit, with a few obvious differences such as the number of tubes being driven,

Converting weight into a proportional electrical signal is also well understood. Load cells are commonplace and can be gotten in many weight ranges and many configurations from a multitude of sources. A load cell + an appropriate signal conditioning chip (amplifier) can give you an analog voltage proportional to weight. Or add an ADC and get the weight as numbers over I2C. Take a look at what this guy did:
learnarduinonow.com/2015/06/06/arduino-hx711-digital-scale.html
I'm not implying that his way is the best way but it gives you a general idea of one approach.

So... if we have a means of driving the tubes from the micro and a means of getting weight sensed and fed into the micro, it's just software job from there. Probably pretty easy software since a scale isn't going to need the user interface and tons of features that something like a clock needs to have. Just read the weight signal, maybe apply a little smoothing and/or scaling and/or calibration math, and send it out to the tubes. Lather, rinse, repeat, 10 times per second or so. Any Arduino, or a "naked" Atmega or STM-32 or PIC chip can handle that without even breathing hard.

So in my mind the electronics and programming is really just combing a few things that have been worked out long ago by other people. Thus "mechanical problem". To me, the hard part would be designing a scale housing and weigh platform that holds the load cell properly, transfers the weight from the platform (the thing you stand on - I'm only assuming it is called the platform) to the load cell, etc. Load cells are fairly sturdy but your housing would need to constrain motion such that the weight is applied to the load cell reasonably "on axis" e.g. sideways forces on the load cell cause inaccurate measurements. Excessive "off axis" forces (sideways, twisting, etc.) might actually be able to break the load cell.

I am not a load cell expert but I have worked with them before. My experience was industrial with load cells working in the 1 - 10 tonne range, but load cells in stuff like butchers' scales work the same way, just smaller.

To whoever else is reading this: if you have a different idea or can expand on my idea, please speak up! Don't let that Admin tag by my name put you off - I'm quite willing to listen to your ideas.

Also this:
en.wikipedia.org/wiki/Load_cell
Strain gauge type load cells are what I'm talking about.

That's a lot of questions Been thinking about it, I'll try to answer at least some of them.

First - battery. Trying to run the whole thing off of batteries is unrealistic unless you have a really big battery. You really ought to plan on using a "wall wart" to power it. If there is something you need it to remember through power interruption (calibration constants?) you can run the microcontroller in an interruptible sleep mode for a long time off a coin cell. Or simply have your code write the things you need it to remember into EEPROM whenever they change.

As an aside, several of my own clock designs use the coin cell method to keep time (in the controller, a 328P) through power outages, The designs pre-date things like ESP-8266 that now make NTP sync over WiFi easy, and if the Dallas RTC-on-chip existed then I didn't know about it. Anyhow, the 2032 coin cell keeps the 328P powered well enough to sleep and wake up once per second to increment the time. Potentially the battery lasts a very long time (never really tested it to extremes).

Driving the tubes. You can learn a lot on this subject by studying the schematic and analysing the code of one of Ian's open source clocks, You can make the high voltage using Ian's technique of using the 328P to control a basic boost circuit, Or you can get a MAX-1771 based HV power module cheap from a number of sources. I;d recommend using a 1771 based module but YMMV.

Most of the other tube driving stuff is already worked out for you - just look at Ian's open source clock schematic and code to see how to drive 6 tubes then scale that down to 4 tubes. That will get you on track for making a multiplexed display scale using one 74141, four optos, etc, If you want direct (non-multiplexed) drive, forget the 74141 and optos, and use two Microchip (nee Supertex) 5530s instead. 5530s are my method of choice but again YMMV.

I'd use an Atmega328P-PU as the controller, but that's mainly because I know it pretty well. It's a well established chip (it's the heart of Arduino UNO these days) and it's stable in my experience.

I just realized that I'm assuming you are planning to design your own board for this project. If I'm wrong, and you really want a kit (as the topic of the thread actually says!) then most of the above does not apply,,, oops. As far as shields and other ready to run stuff, I don't know very much. I'm not aware of a shield that runs 4 tubes but that doesn't mean one does not exist. "Shield" implies "Arduino". Except for the backup battery stuff, which may not even be necessary, I guess you could do it with an Arduino and some sort of shield. That's not a subject I'm well versed in - I do this stuff for fun, not for business, and I enjoy the challenges of designing from scratch and getting boards made rather than using Arduinos and shields.