Several years ago I rebuilt my 82 Odyssey Engine, but it turns out the connecting rod is too long. I now find that a rod kit is available, but comes with a cylinder base spacer to account for the long rod. Can anyone tell me how thick the spacer needs to be, or tell me how high the piston edge should come at TDC (Top Dead Center)

I don't have the answer you are looking for but I can point you in the right direction, see if I get this right, someone is sure to correct me, here goes.
at top dead center the top edge of the piston will be very close to the top edge of the cylinder with the piston at BDC, the top of the piston should be even with the bottom of the exhaust and transfer ports. the head gasket and head squish area should give the clearance you need.

The correct length rod is no longer available?

Usually when a longer rod is use they just make a spacer the same length as the longer rod, if the rod was 5mm longer use a 5mm spacer.

Unfortunately I didnt know the rod was longer, and don't have the old rod now. I have no idea how much too long the rod is.

At one point I added an extra cylinder base gasket, but it wasn't enough. The compression was still so high that my goki starter would barely turn it over.

64.4 mm (stroke) from the bottom of the intake ports puts the top edge of the piston side at 2.8 mm from the top of the cylinder sidewall at TDC (Top Dead Center)

You need to know the difference in the rod length then raise the cylinder that amount, knowing the stroke does not help.

You have no part number of the rod used or know what other Engine the rod you used was for?

I'm not sure if there may be a part number on the rod or not. I will look!

I doubt you will find a part number on the rod you need the part number off the box of the parts you installed too often guys are too quick to throw this stuff away and never document anything or read instructions, no big deal take the Engine apart order the correct rod and do it all over again this time take measurements and read.

Tried to press apart a old FL250 crank here to see what the length of the rod actually is and to see if the rod kit I have here would work, my press is not strong enough to press it apart, sorry this crank will have to be sent to Speedchaser his press is stronger, not sure when that is going to happen, also have a 350 crank for him to rebuild.

I stand corrected look at the side of your rod for a number PROX does put the kit part number on the rod at least they did for this CR500 rod kit.

Maybe we can get lucky here.