Set it up on a separate percussion track?

This will do what you want

It worked!  But now another issue: The composition is in 4/4 with a quarter-note pickup. It gives me 4 clicks, followed by 3 counts of silence, then the pickup note.

Can I get it to give me 3 clicks, then the pickup note?

… Can I get it to give me 3 clicks, then the pickup note? …

I could be wrong, but I suspect that the number of clicks must be an entire measure, in the current time signature.

That is: 4 clicks when the time signature is 4/4.

Hence:

1) Do not use the Finale feature Pickup Measure.

Go to

Document menu > Pickup Measure…

and clear the pickup.

2) Time Signature Tool.

Set up the first measure as a 1/4 measure, displaying a time signature of 4/4.

3) In the dialog Click and Countoff, set the Countoff to 3 measures.

Then the Countoff will be 3 measures of 1/4.

It works!  But now I tried exporting it as a wav file, only to find the countoff clicks are not audible. Is there a fix?

If you want the clicks to export, then you have to built that into a percussion track (as Mike R suggested).

You can always remove the extra before printing (or keep separate files) if you wish.