maybe i can help by explainging the system and how to possibly pin this down. because if this is a good battery then you may have either a faulty connection, faulty starter relay, or the starter brushes may need to be replaced.
a good place to check all this is at the starter relay. if you listen for that click noise you get when you press the button, you can find that relay.
on that relay there is two larger posts. one of the larger wires is the positive battery cable comeing from the battery. the other is the cable that goes to the starter motor.
first off check all these connections make sure they are tight. the good thing is that the starter clicks, meaning the starter is getting signal from the starter switch. that relay just makes the connection between the battey cable to the starter cable. so if the starter motor was in good shape and the battery was in good shape, you can cross or jump those two larger posts together with a screwdriver or something and it will turn the starter motor over. if this is the case the starter relay is bad because its not makeing a good connection.
so if you jump those two and it does not turn over the motor then this means the battery is at fault or the starter. so if it is determined that the battery is good then this just leaves the starter motor. and usually to fix that it just takes a set of starter brushes to rebuild it to a good working condition.
again it would be good to know that the chargeing system is up to par, but maybe out of what i wrote above you can diagnose the system according to what you are running into.
hope this helps. thank you.