Unfortunately, the only way to find out where the slop is to take it apart.
Since the play is axial as opposed to in and out, I would suspect gear wear. The side gear adjusters are fixed in place with a locking tab, so I don't think the ring gear has somehow backed away from the pinion gear. That leaves plain 'ol surface wear on the pinion and/or ring gear.
I think the dealer is trying to gently brace you for a bite out of your wallet.
The only way to avoid the dealer cost of repairing the differential is to take it apart and fixing it yourself/having a mechanic experienced in differential work fix it for you. Working on and setting up a rebuilt differential is a tedious chore even when you know what you are doing! That is what makes it a pricey repair.
For an interesting tutorial on differential rebuilding, check out this link:
Samurai to Sidekick gear swap. - Zukikrawlers
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