Regarding the wandering zero problem to quote Supertramp,
"Now some they do and some they don't
And some you just can't tell
And some they will and some they won't
With some it's just as well" From Goodbye Stranger.
More seriously, the Enfield is a pretty complicated rifle to accurize as its design is not ideal for such. The additional lightening cuts, the barrel, the rear locking action, the separate butt and forend, all make things a bit complicated along the pedestrian accuracy of Enfields to begin with. It didn't help that they used a wood stock in the jungle either with multiple points for water entry. FYI,
http://leeenfieldresource.com/?page_id=21 Roger Wadham goes more into depth in his book, the 2012 Complete Book on Lee Enfield Accurizing. I've been rebuilding a No. 1 using his book with a new old stock barrel that I acquired.
I suspect that unless you fired using a clamped rifle in a benchrest remotely and fired at ranges over 100 yards, it would be difficult to detect. Some folks that reported a wandering zero tapped the jungle carbine and used a scope. It could just as well been crappy scope mounts, the significant recoil, and the less than perfect scopes of yesteryear. Also, a fair amount of these are actually faux carbines and depend on the craftsmanship of those who did them. All this plays into the maybe it does or maybe it don't. It is undoubtable though that what the Brits did did not enhance the accuracy of these rifles--they were for a quick and dirty job with minimal investment in research, design, and tooling to do so.
However, these have now become pretty expensive to use as a knock around truck gun if they are the genuine article. Condition is everything to a collector. The rubber butt pads on these, if original, won't last under sustained use and collectors want the real article--not an aftermarket replacement. That is why I recommended finding one of the fakes carbines like a Santa Fe, or shortened sporter barrelled No. 4's or No. 1's,, and if one's heart is set on a fake carbine, then cutting off the crappy aftermarket flash hider and putting a muzzle brake would help tame the recoil. I am not particularly affected that much by recoil on long rifles but I find the Jungle Carbine with regular loads to be unpleasant due to recoil and blast. I don't like Mosin carbines for that reason. And I am enough a purist that I won't put a muzzle brake on these just to shoot them.
I like the K31 and also have the K11 which is pretty similar. They are awesomely accurate rifles compared with most run of the mill milsurps; in part because the Swiss did the issue ammo right and kept their rifles up. Occasionally, you will run across a sportered version of these and Swiss Products does make a muzzle brake just for the K31. As a collectible, I suspect that these will continue to increase in value.
FWIW, I suspect that folks that actually hunt bear might suggest different choices than military surplus rifles. If it is just for self defense against bears, as said above, a twelve gauge repeater with slugs might be the easiest choice.