What about some 3mm ply strips?
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using plywood strips is possibly a lot more work than using builders' bog
is 3mm ply actually 3mm thick or is 3mm ply actually 3.175 mm -- aka 1/8" ply ?
and if the track saw cuts a kerf thinner than 1/8" -- I understand that Festool blades leave a 2.2 mm kerf
bit narrow to back fill with 3 mm ply
As you say, Ryan, your Dad's railway was not designed to me moved, to be disassembled and reassembled.
I am concerned that once you start cutting rails and cutting the "landscape" you will never be able to realign it properly again. Cuts are never perfect and errors always compound.
Have you considered unscrewing all components - rails, turntable, bridges, whatever, to protect them from damage and cutting.
Next cut the landscape into movable pieces.
Then reassemble the landscape at the new venue, assuming that it will physically fit. If too much damage is done, it won't realign adequately or it will not fit, then rebuild the landscape. Start again with the old hardware. I bet your Dad has plenty of ideas!
This will lower risks, and probably reduce the work needed.