Instead of a step-down converter, you could use a negative voltage regulator like the LM337 to generate what will become a common ground for the ESP32 and HC595.
For the temporary 5V supply, perhaps you could power the HC595 at 5V. That would still let you drive the transistors, but the ESP32 at 3.3V might have trouble driving the 595. Actually, that would probably work, but just isn't guaranteed.
I'm not sure if there's a switching equivalent of the 337.
With these you can use TPIC6B595 instead of 74HC595. Then you will not need any transistors. You will need current limiting resistors for each segment, and a supply voltage ~2V higher than the segment forward voltage.
With TPIC6B595 there is a problem. These chips need 5V supply, they won't run at 3.3V. That means you will need to boost the 3.3V signals from the ESP up to 5V. A 74HC14 would do this.
You might look at the TLC6C598 instead. It will run on 3.3V or 5V, but the minimum VIH is a fixed 2.4V regardless of Vcc, so you should be able to drive it directly from the ESP32. It will do 50mA per line continuous when Vcc = 5V. Presumably less at 3.3V. It's a 16-pin chip. The datasheet shows it available only in TSSOP, but Digikey carries a SOIP version, which could be the automotive version.
I can't find any information on the maximum continuous current per segment on either display. Anybody have better luck?
I understand the preference for through-hole. But could you use an SOIC-to-DIP adapter board? You would have to solder the chip to the adapter, and solder the header pins. But then you have something that plugs into a breadboard.
I will need to check how small it would be, I only got a soldering iron.
And I can't find much more about the displays too, expect the stuff they say in the description. But I can post the model number written on the part tomorrow
For common anode, another chip to look at is the TLC5916. It is 16 pins, and comes in the through-hole DIP package, and has constant current control using a single resistor. So you wouldn't even need the 7 segment resistors, and brightness would be easily adjustable. There are several Youtube videos on it. Looks very promising.