otters are deemed to be 'native', whilst barbel/carp are not... well, what's the problem? It may be a cynical view but I believe the word 'native' is used here as not much more than a device to justify one (cute) species being given a free hand/paw to destroy another (non-cute) species. Cute and 'native' ... what's not to like eh.
I think people confuse "native" and "indigenous". Barbel are native to the country as a whole. However, they are not considered to be indigenous to rivers such as the Ribble, Severn, Wye and Southern chalk streams etc. The EA appear to have become largely indifferent to the barbel populations in such rivers. Where a barbel population is not indigenous to a particular river the EA seem to be unlikely to do much to support that barbel population. Only rivers emptying into the sea on our eastern shore seem to be considered to contain indigenous barbel populations. This may not make sense when you consider that EA predecessors often had a lot to do with some initial introductions on these rivers. However, policies change over time, such is the way of big government agencies. Similar policy changes have also impacted the stocking of other species. I know of waters that got licences to stock cats and sturgeon many years ago. If they were to try to stock more now, they'd be refused out of hand.
Carp are an oddball. They are not indigenous to this country at all. However, they have been here so long that they are considered to be native. Regardless of that, they'll never be indigenous. The same does not apply to more recent introductions such as zander and wels catfish, both of which are not indigenous yet are still considered to be invasive species despite being first introduced well over one hundred years ago. There does indeed appear to be a time scale applied. How that works is anyone's guess. It may also change with different people making the decisions. I doubt it is set in stone.
Whether we like it or not, otters as a species are both native and indigenous to UK waters. The fact that they were pretty much wiped out in many areas is utterly irrelevant.