Think hard about whether you should keep it.
(In the above, there's the main entry story, and in one of the comments there's a second story.)
This next one does _not_ involve damage to a house as a result of a dog fence/lightning combination, but does describe damage to the fence during lightning storms. Twice. Lifetime warranties on the fences seem to be good for covering these. Further in the article, the "expert" chuckles at the idea that a surge protector would do any good at all.
I've heard of something similar happening here in town, which is what prompted the googling.
The wording on the FAQs for the products is suspiciously careful when answering the is-it-safe-during-a-lightning-storm question. The internal and/or external surge protectors are plugged (heh); often there's a lifetime warranty on various specified system parts if damaged by lightning. But there's no discussion of whether, in the event of lightning being channeled to your home and your home being damaged, the makers of the product will pony up to fix the problem.
Think really hard. And maybe talk to whoever you get your homeowner's insurance from. They might have an opinion about the safety of that invisible fence -- and whether they'd exclude coverage of resulting fires.
In the meantime, here's a really entertaining site you can waste time at:
You really gotta feel for the people who had a car accident and then had a near-hit. Geez. Not their lucky day. Or maybe it was -- maybe they should buy a lottery ticket. Neither of the two events killed them.