Scooterotica

I have just charged the battery on my GTS 300, however I did not realise it is not advised to use a car battery charger. Although the battery now seems to start the scoot ok, have I done permanent damage to it?

Not being familiar with the GTS 300 I can only offer a general opinion.
It's probably unwise to charge the battery without disconnecting it from the scooter wiring, as chargers usually put out over 12v (typically 13.5v)
Smaller batteries are best slowly charged rather than rapidly. A big ol' car charger MAY pump the voltage in a bit quick.
However, if the battery has survived and is working, I wouldn't be worried about it.

You wont hurt your scooter OR battery using a car charger, thgey are almost all intelegint chargers these days and reduce the chargeing current as the batter nears full charge. It is best practice to didsconnect the positive side of the battery when charging but I seldom do and havn't had any problems in 50 years of motoring and 10 with the GTS