When deciding your course of action, be aware that the Shimano Dura-Ace SL-BS77 9-speed bar-end shifters are not very Dura. The critical Amazon US reviews seem to have the most information about this design flaw. (The positive reviews everywhere are written a week after buying the part.)
On the other hand, the 9-speed down-tube shifters are highly reliable....
what is weird is that (as I understand it) the guts of both levers are basically the same. What I do know is that you cannot install the bar end levers without removing the lever and its bezel from the mount. The bezel ( a chrome plated zinc die casting) looks as if it will fit several different ways on the lever mount but in reality it will only fit on one way.
If you fit it wrongly, after a period of time the bezel will break and the lever will fail to work at all. At least some of the claimed 'design flaws' are undoubtedly due to bad installation. If the bezel fails and there is no other damage then a new bezel can be bought for a few quid. Talk of there being 'no spares available' is not quite correct.
I would like to get my hands of a set of failed levers to see if there is more to it than this (there might be). But so far no-one has come up with any for me to look at.
I've killed several sets of these shifters by mounting them on an USS recumbent bicycle (which means - in the absence of a sacrificial bell - the shifter is what hits the ground if the bike falls over, and is what the bike rests on if you lean it against a wall without first shifting to a position where the lever isn't sticking upwards).
The failure mode seems to be that the indexing detents are achieved by a cheesy plastic ring internal to the shifter, which cracks under pressure. This keeps mostly working for a while, doing a passable impression of a gunky cable, but eventually spreads apart and the indexing goes to pot. The lever generally remains usable in friction mode (though I did have a crash-damaged one that eventually lost its friction due to more extensive internal damage).
Since bar-end shifters are more commonly installed facing downwards on drop bars, the chances of mechanical damage are much lower. Incorrect installation of the bezel causing pressure on the mechanism certainly seems plausible, if the installer hasn't read the manual or noticed that it's not symmetrical.
Anyway, after three rounds of replacements, I tried a set of
Microshift BS-A09 instead. These lasted several years of bike-falling-over-due-to-kickstand-sinking-into-muddy-ground and train tetris without problems, until I lost traction on some chutney and landed hard on the rear shifter (which, to its credit, still managed to index, even though it had become stiff and lost much of its aluminium to the road). The A09 doesn't have the option of switching to friction mode, but my experience of the Shimano is that that won't necessarily save you long term if the problem is the shifter (rather than, say, a bent derailleur hanger). Unlike the Shimano, the mounting bolt is responsible for adjusting the friction of the lever (there is no adjustment on the Shimano). Subtlety and Loctite are required to achieve optimal feel and shifting performance - if you're gung-ho and overtighten it, the lever will stop where you release it rather than settling in a consistent position above the detent, and the indexing won't work properly.