I work on stoves every day and see lots of veneer stone walls. And yes it does fail sometimes even when done right. It can even with lath. Behind a stove it is exposed to fast and large temperature swings which will be even worse for the op because it isva cabin. To me the little bit of extra money and time is worth the extra insurance.
Like I said, I'm normally a fan of overkill (belt and suspenders approach). But putting lath under the cultured stone can make it worse, not better (because the thermal coefficient of expansion of metal lath is many times that of the under-lying cement board). Using either installation method requires that the substrate (cement board) be solidly supported and rigid or the install will fail. The definition of "doing it right" is it will not fail in normal use. Normal use includes wide temperature swings. Most failures will be due to one of three failures to "do it right", not whether lath is used or not:
1) Cement board improperly installed (not supported or affixed in a manner that makes it rigid). Extra blocking is good insurance regardless of method used.
2) Cement board sucks too much moisture from mortar.
3) Mortar not mixed to proper consistency or not applied within the allotted time.
Adding lath will not prevent any of these issues although it could reduce the impact of #2. If you use lath and have issue #2, you better have affixed that lath to the cement board like your life depended upon it or it too will fail. Pre-dampening the cement board is super easy compared to adding lath and doing two applications of mortar. Lath had it's place when plaster was popular (and is still required for many exterior applications) but acrylic latex modified mortars are a completely different animal compared to plaster.
For a woodstove surround I would spend extra money on a mortar that is highly modified with an acrylic latex admixture. This will greatly improve the ability of the install to handle thermal stresses (lath will not). You can either buy a highly modified mortar or you can use a liquid acrylic latex admixture made for this purpose. While highly modified mortars and latex additives do cost more, they don't increase the amount of mortar needed (as lath does) so the overall cost will be the same or less (while the labor will be greatly reduced and the durability greatly improved). Mortar is a general term for a wide variety of cement based products. Thinset is a type of mortar and would be fine here if you are going for a uniform "flat" appearance (but it's not suitable for applying in thicker layers). Being a "belt and suspenders" type of guy, I would buy a quality acrylic latex thinset and fortify it with additional acrylic latex liquid admixture to insure it handles the thermal stresses. And, yes, pre-dampen the cement board. This will make all the difference in the world.
Regardless of lath or direct install, it's critical that proper mortar mixing procedures are used and time windows are adhered to. Mix in small batches, allow the freshly mixed mortar to slake for the specified time (5-10 minutes) and throw out any mortar that cannot be used in the specified time window. Starting with very cold water can help extend the time window a little bit.
Properly installed it will not fail. Lath adds a lot of hard labor with no benefit.