I think your HV and filament power needs are met by the DC-DC converter (I see the filament traces routed to the multi-winding inductor).
It looks like a four VFD PCB with the bottom one not fitted. What is interesting is the multiplex layout. It looks like first (top to bottom) pair of VFD have anodes connected and independent to the next pair of VFD anodes. The grids from the 1st and 3rd are connected and independent of the 2nd and 4th VFDs. 35 anodes (1st, 2nd VFDs) + 35 anodes (3rd, 4th VFDs) + 20 grids (1st, 3rd VFDs) + 20 grids (2nd, 4th VFDs) = 110 bits needed which is below 120 bits provided by the 6 (20 bit) shift register chips. So the multiplex will light one digit from the upper pair and one digit in the lower pair per 15 bytes of data and LS load pulse.
Can you see pins 1 and 30 of one of the shift register chips?
If they are open, then the devices are in serial mode (not T-BUS).
The LS, CLK and DIN lines are common between the 3 modules.
Are you sure or is the DO(pin 2) of one chip connected to the DIN (pin 29) of the next chip?
Can you trace LS, CLK, DIN, to the PCB connector?
I see 7 thin traces going under the PCB connector. Apart from these three, there CG/DO and /CL plus two more unknown pins. DO and /CL can be left open as it is an output or is pulled-up (to a functioning state), respectively.
If you power up as shown (just the power pins) does the display show any life (ie. dots lit or brief flashes)?
You can create a program for your Arduino to send out 15 bytes (all high) then toggle the LS pin (high then back to low) and see what lights up. Just in case, use 1K resistors between Arduino pins SCK, MOSI, SS to this PCBs CLK, DIN, LS, respectively.
For quick and dirty, you can just connect the DIN through a 1K to high, LS through a 1K to low, toggle the CLK >120 times (555 osc), then pulse LS high. See what happens.