I have a 94 TA since 95 and have known many with TA´s and AT´s during the years since then. Been on Transalp lists since way back, forums. Deti´s Transalp Pages, Transalp.de etc.
I cant remember ever having heard of this problem at all, something must have been changed, parts missing, wrong type or incorrectly installed is my first thought. This should not be an issue on a normal Transalp. CDI´s on pre 94 models are known to fail, rear wheel bearings wear out early on many, speedo drive gear often fail and on early models; I believe 87-88 starts to use oil at some point. Apart from that, bomb proof bike if normally looked after.
I looked at the pictures; your chain is way too tight if it is on it´s side stand. If you have ridden with a tight chain you should both feel and hear it easily and that will most likely cause premature wear on parts. it says 3,5-4,5 cm chain slack, 3,5 is actually too tight, stay at 4,5, its an off road bike. Better a bit loose than a bit tight.
You say it is an 88 model? Maybe the engine is but the rest of it is not. Can that explain anything irregular? It looks as if you have an early Transalp engine (black) installed in a 97-99 Chassis. Double front discs, swing arm with rear disc brake, fairing is 94 or newer. Silver coloured rims (96->) etc. Do you mean you have a 98 TA with an 88 engine?
Its a strange problem, maybe a mismatch between the Transalp parts you have there?
Anyway, good luck!