I have couple of questions:

Do the flashers and coils work perfectly with incandescent flasher bulbs?

What make/model LED flashers are you using?... Are they 13VDC rated?

On a curious note: the red, yellow and blue mars flasher pairs each have a 600 ohm warming resistor. The other 5 pairs of flashers which are multiplexed with regular coils do not, just a 5 ohm current limiting resistor. I'm not sure why that is, and of course schematics could be wrong (gee, that never happens!) Each are parallel wired sockets. Make certain that you always replace them in pairs and and do not mix with incandescent lamps or you will get undesirable results.

That said, according to the Laser War club on Pinside it has been noted that someone has replaced the flashers with LEDs successfully, in the first post.
https://pinside.com/pinball/forum/topic/laser-war-club

The remedy for F14 is to remove the warming resistors, unfortunately the DE Laser War schematic only shows a few of the flashers as having warming resistors, most of them do not...I'm not sure why. From the Laser War club thread, it seems that someone has used these without modification - perhaps try to send them a PM.

I would really want to confirm the LEDs are the problem. A local auto parts store should have some 89s.