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Absolutely understand not wanting to touch carbs. Getting to them can be a pain as well.
In fairness to the seller, bike batteries are relatively shit. Batteries on 125cc are especially shit. My old 125cc had a rubbish battery.
In fairness I replaced it with another cheapo, as I reasoned using a battery tender and replacing it relatively regularly was economically proportionate to its value.
For my Triumph twin I splerged on a higher cranking on - but also because the EFI are notorious for needed more amps(?).
If the battery is aok, and increasing the fuel (ie throttle open) keeps it running then it sounds like it may be something to do with the air / fuel mix.
My order of checking would be:
At as an interim 3. and 4. could be achieved with a stiff brush and spraying a shitload of carb cleaner spray down there. If it is the carbs and they need a proper clean then that's a bit of methodical time and patience (and heavy note and photo taking)
lexomotos will be a cloan of a jap engine. He should find out which one it is.