It's been a while since I saw this one!
Your Solid State Flipper Board 520-5070-00 has a problem. I'm currently at one in five replacement rottendog solid state flipper boards having problems when brand new.
The schematics for your game are here:
https://www.ipdb.org/files/1416/Data_East_1993_Last_Action_Hero_Schematics_BW.pdf
PDF page 20 has the schematic for the Solid State Flipper Board.
As always with ANY problem with Data East flippers of this age, get yourself Qty 3 180-5124-00 End of Stroke Switches, and if you don't use them, throw them in the bottom of the game. You'll need them eventually, and when I have two Data East flippers, I frequently by the end of the troubleshooting have used three new EOS switches.
This flipper design can be a battle. The EOS switches cause a flipper to fail to activate, so it's possible that your problem is EOS switch related... but this specific problem for me was a bad flipper board.
In your situation, I'd do this:
Install the Rottendog board you have.
Take an alligator clip, and clip one end on a switch terminal for your right flipper EOS. Use the other end of the alligator clip to clip to the other terminal of that same EOS switch. This shorts out the EOS switch, and makes SURE that it is registering as a solidly closed switch to the Solid State Flipper board.
Test your flippers. Make sure the LED's light up perfectly on the board when the flipper buttons on the side of the cabinet are pressed.
If the flipper starts working perfectly when the EOS is bypassed, I don't fiddle with that EOS switch anymore. I replace it, and adjust the new switch VERY CAREFULLY, to be very, very tightly closed, and when the flipper bat is raised to have an opening about the thickness of a dime between the contacts.
It's handy to have a switch adjustment tool for this, because the blades on these EOS switches cannot be mangled, and your typical needle nosed pliers will mangle the switch.
https://www.marcospecialties.com/control/keywordsearch?SEARCH_STRING=switch+adjuster
Some facts about your Solid State Flipper board. The board will work fine with the alligator clips across the EOS switch. But when the ball hits a flipper bat that is raised, it will knock the bat down, and the game won't know, and won't raise the flipper bat back up. So alligator clipping across the EOS switches will restore operation to a flipper (provided your board is good), but not proper operation. I've left the alligator clips in place while I wait for replacement EOS switches to arrive.
My board was bad, and caused this symptom. As I remember it, I strongly believed that I had to have some other problem because my Rottendog board was new... and it was a bad Rottendog board.
I'd double down on the cheaper possibility, that you have a maddeningly common Data East EOS problem, but if you still have problems with the EOS switches bypassed, and the flipper buttons are lighting LED's on the Solid State Flipper board, suspect that you've got a bad board.