Praise be to Allah.
On our website we have a number of fatwas in which we explained that it is not permissible to bow when greeting someone, whether that is in the context of sports or martial arts, or otherwise. All of that comes under the heading of that which is not allowed according to Islamic teachings. We also quoted the evidence for that being disallowed. See these fatwas under the following numbers: 20198, 121638, 4259.
If bowing in greeting is optional – as the trainer told you – then there is nothing wrong with you joining these classes and taking part in the contests and training, and offering your excuses to the trainer for not bowing, out of respect for Islamic teachings, and greeting with a handshake instead of bowing.
With regard to striking the palm with the fist of the other hand, there is nothing wrong with that, so long as it is done on the basis of custom, whether it is a social custom or a custom in that sport. The gate of custom is wide open, and customs are permissible and not restricted in general terms.
This sport has become global and does not belong to any particular nation to the exclusion of others, and among the Muslim peoples there are many kinds of greetings, such as touching noses, patting the shoulders, and so on, and none of the scholars have said that these customs are prohibited, so long as the Sunnah of greeting with salaams and shaking hands is also still observed and has not been replaced entirely by such customs, praise be to Allah.