Resolved.
Apparently while all info points to only doable once, you can actually repeat it - at least you can on different factions (that’s how I completed it on the next try).
So don’t be stupid (like me) and have a full pet log before doing this “quest” - make sure you have room. Jenafur will not go to your mailbox or to your bags if you don’t have room. You just get a completed (hidden) quest without a result.
Other posts everywhere have stated (even by blues) to contact CS directly if you have this occur; however, it seems to be that not all GMs are up to par on this information.
Since there is no actual official “quest” to be stuck on, I’m not sure the “stuck on quest” ticket applies; in this case, it’s a loot issue because the reward is given to you automatically. Either way, the GM had no idea what I was talking about.
If you DO get this issue, simply state, “Unable to receive Jenafur due to full pet storage.” That was the advice given directly by blues… except again, the responding GM to my case had no idea what I was talking about.
So it appears it’s only doable once per character, not necessarily once per account, but you can’t have more than 1 Jenafur and they’re not cageable. (I’m just stating some of this extra info in the off-chance that someone else like me comes along, desperately looking for more information than is currently available on this particular issue…)