first of all, thank you for the fast answer,
I was thinking about trying another battery, but I cannot find a suitable one easily. Moreover, the exact replacement (that would be the best choice to avoid doing more damage) appears to be made by Shenzhen cyclelong company, at least this is the only one I found, and anyway cannot be bought as an end-user:
szcyclelong.com/zhan.asp?id=9 model 404343, 760 mA nominal
if you could send me a replacement I would be more happy, in any case, the battery I received didn’t work correctly from start, so I think it can be replaced “under warranty” 
any other suggestions? I tried to test the charge without battery, but I think I’ll need a “battery simulator circuit” to get consistent results, since just putting a small resistor, doesn’t work…