Honestly, depending on what tools you have/will have to buy to do either, it may be easier to rebuild the monitor chassis. However, You do not need to modify the original wiring in any way to put an LCD in your game. The hardest part will be building/modifying brackets to get a LCD to mount securely in the game.
This converter board takes the RGB video signal designed for the CRT and converts it to VGA for use with a VGA compatible LCD. It comes with a pigtail that has all the wire colors you need. Setup is stupid simple, unplug the video pigtail going to your CRT monitor, find out what wire colors on Birdie King correspond to Red, Green, Blue, Ground and Sync. Then wire the Red, Green, Blue, Black and White wires on the converter board pigtail to those wires respectively. Preferably, make an adapter that will keep from damaging the original video connector, but if you cant. I would recommend 18-22 gauge red "Quick Splices" so that you can tap into the original wiring without cutting the original connector off.
https://www.amazon.com/Connectors-Solderless-Electrical-Insulated-Terminal/dp/B0BN14J6NJ
The power is stupid simple too, the converter board will run on anything between 5 and 12 volts DC. Any cheapo barrel plug power adapter with center positive polarity will work in that voltage range. Alternatively, there is a pigtail that comes with the board that has pre-stripped red and black ends, that can also be wired to any 5-12 volt DC source.
Try and maintain the ideology of preserving what is already there, so you can easily convert it back if you want (or for the next poor sap down the road!)
Good luck and feel free to PM or post here with any questions.