You can't swap boards and expect the game to work, as you may be aware.
The majority of the 32 DIP switches are game specific.
The ROMS are different as are the jumpers needed to get the game working.

They are both -35 type mpu's so with some work it is possible to fix both mpu's as imo they don't look too bad, seen much worse.

Lots of information here :https://pinwiki.com/wiki/index.php?title=Bally/Stern

the leaking battery acid damage will continue to spread on both boards unless tended to.

+1 to Weebly

Are you up to testing the associated transistors on the auxiliary lamp driver board and if found faulty then replace them?
Q6, Q1 & Q12.
Have you tried reseating the connectors on the auxiliary lamp driver board?
Have you removed the auxiliary lamp driver board and checked header pins for dry/cracked solder joints?

What happens when you do the lamp test via the button inside the front door?
Do you hear a clicking of the solenoid expander board relay under the playfield?

Maybe consider a new aftermarket auxiliary and lamp driver board:

https://nvram.weebly.com/bally-stern-ldb.html
https://nvram.weebly.com/bally-aux-ldb--43.html

or
https://allteksystems.com/collections/pinball-replacement-board-products/products/auxiliary-led-lamp-driver-board-for-as-2518-52
https://allteksystems.com/collections/pinball-replacement-board-products/products/ultimate-led-lamp-driver-board

Not that this is causing the issues, usually a lamp won't light if there is a dry solder joint, but I definitely see dry joints on both connectors.

A lengthy read but very informative read: https://www.pinwiki.com/wiki/index.php?title=Bally/Stern

Regarding the clicking of a relay....I misinformed you, sorry.
The solenoid expander board relay under the playfield is driven by the Q11 transistor on the solenoid driver board, not the lamp driver like in other games.

Before changing any components I suggest reflowing your header pins with solder.

Even if you do swap out components those dry solder joints need to be addressed.

So you have a soldering iron station or even a cheapy, and some solder just to try out on your auxiliary driver board headers?

For learning, this situation is ideal imo.

You would just be heating up the solder joined to the pin until you see it melt and add a little solder, remove your soldering tip from the pad/pin and let it cool.

Again, I don't believe a 'stuck on' light is the result of a dry solder joint, but where you have lights that are intermittent it should fix it.

For example, the right hand side bottom pin and top pin you can see a circle in the solder around the pin, as in the left hand side 4th pin from the bottom, there are more.

That circle is a break in the solder connecting to the pin, there must be solid contact between the solder pad, solder and the pin.

These 'dry joints' can occur from the header pin being moved/having pressure put on repeatedly from removing the connector on the other side.

There are dry joints on the second and third pins from the bottom on the right hand side, but these are not used as the don't have a trace connected to them.

Where you don't see a trace connected to a pin is where I suggest you try out repairing the dry joints.
This way you will see what is going on but don't have to worry about applying too much heat/solder as it won't ruin the trace pad, if that makes sense?

Upon applying the solder tip to the solder and pin you will see the solder change in colour slightly as it melts.
Adding a little fresh solder, it will 'join' to what is there, ideally creating a new fresh join.
As the new solder is joined you remove the soldering tip from area and you're done.
Depending on the solder you have you may notice fumes coming off, this is flux that helps the metals join, ideally don't breathe it in, also it may leave a brownish residue there, this can be removed when cool with methylated spirits.

regarding the flickering......are there original incandescent globes or LED's installed?

in the pic in #4 of the 3 light sockets, the ones remaining on.....can you please upload close up pics of lamp sockets showing the wire connections to the sockets and the braid to the sockets?

it's probably just the pic but it looks like the ground wires are touching the bus wire?

also you could remove the globes and using your multi meter check for resistance/continuity between the two tabs on each socket.

if you are unfamiliar with using a multi meter to check for continuity/resistance please upload a pic of your multi meter (if you have one) and I'll explain.

you already reflowed the headers, when you tried fixing the dry solder joints, the separate solder joints on the rear connect to what is called a 'header', the male part with the pins in a row.

the female part is the connector with the wires attached, in your case IDC's (insulation displacement connector)