Actually almost all such Eastern martial arts, or Yoga and Meditation and etc. have background philosophies based on atheistic and non Heavenly religions, but there would be no problem for you to learn and practice the arts as long as you don't accept their philosophies and ideas, and look at it as merely a body treatment and sport. Even you can set your intention to be something like "
I'll exercise to get ready for Jihad and fighting for the religion of Allah, or for defending myself and my religious brothers and sisters" [in today wars the body strength is not as important as in the previous centuries if you are not a specialist soldier, but defending against robbers and etc. is still a good reason, even becoming healthy as thanking Allah for the gift He has given to you is a good intention on its own]. Maybe the following verse can be related to this issue as well(?):
وَأَعِدُّوا لَهُم مَّا اسْتَطَعْتُم مِّن قُوَّةٍ وَمِن رِّبَاطِ
الْخَيْلِ تُرْهِبُونَ بِهِ عَدُوَّ اللَّهِ وَعَدُوَّكُمْ وَآخَرِينَ
مِن دُونِهِمْ لَا تَعْلَمُونَهُمُ اللَّهُ يَعْلَمُهُمْ ۚ وَمَا
تُنفِقُوا مِن شَيْءٍ فِي سَبِيلِ اللَّهِ يُوَفَّ إِلَيْكُمْ وَأَنتُمْ
Against them make ready your strength to the utmost of your power,
including steeds of war, to strike terror into (the hearts of) the
enemies, of Allah and your enemies, and others besides, whom ye may
not know, but whom Allah doth know. Whatever ye shall spend in the
cause of Allah, shall be repaid unto you, and ye shall not be treated
and when the intention is for Allah the deed would be even considered good! In a famous Hadeeth from Imam Reza --peace be upon him-- Imam specifically stresses "الأعمال بالنیات" which implies the worth of the deeds is measured by the intentions behind them. Let me bring you an example. You see a glass of water and drink from it and all of a sudden understand it was actually wine, if you immediately stop drinking that drinking was not Haraam since your intention was drinking water, but if you think there is a glass of wine and start to drink and find out it was nothing more than water it will still be like drinking wine in the view of Allah and thus Haraam and sin, not being submitted to Allah. So that a unique deed can worth differently in the view of religion if their background intentions are different when done by different persons. The only thing that may be not recommended is that you would pay a teacher to learn you the art and this will help the background wrong philosophies to spread and grow in the society. That this is OK or not depends on you yourself if you think it is a relevant reasoning then you should stop it and anything like it, and if you think not then you can pay for it. In the country that I live the same arts are taught almost all by Shia Muslims and no such wrong philosophies are considered at all. So the situation largely depends on you yourself and the class that you want to take part in.
Also bowing in such sports is to show respect only and not worshiping, thus, no major or minor shirk is involved, as long as we are allowed to respect the person (to whom we are expected to show respect) to any extent (I'm not sure but there might be people who are enemies of God and any respect to them is forbidden, and of course no one can be identified as an enemy of Allah unless it is proved).
PS. Let me add two points to this answer for further clarification:
Question: Can a Muslim join a Satanic cult and perform their rituals as long as they do no accept their philosophies and ideas? Answer: Is there any personal or cultural or Islamic profit in doing so? When there is no advantage in doing so and only disadvantages (like helping the wrong ideas to grow and spread) are probable then of course it would be wrong and as a religious rule it would be Haram (forbidden due to the nature of its disadvantages, otherwise it could be also Makrouh, like smoking is based on the ideas of most of the scholars). However, if you found any profit in taking part in their rituals then it could again be discussed, maybe allowed and maybe forbidden. As an example suppose you live in a city that almost all of its citizens are members of a Satanic cult, and you cannot immigrate for some reasons, you should live there and at the same time that you pray the only God you also want to save your life not to be executed by those people. Now the answer to the question would be "Yes, do Taqiyyah!" which means: pretend you are one of those but pray the only one God in your solitude times. This answer has many witnesses in Quran cited elsewhere.
In Shia jurisprudence (fiqh) it is forbidden (Haraam) to appear in the society in the traditions of atheists, like it is Haraam for a man to put on the women's dresses and vice versa. However, you may then ask is it forbidden to wear a tie in public? It depends on the society that you live, is it considered only as a clothing of the atheists or even Muslims commonly use it? So that if the clothing of such arts are considered in your community as clothing of atheists then it would not be allowed to use their clothing while practicing their sports, if the art is considered a ritual practice than a simple sport then again it will be forbidden for you to do it, and this thread can be continued more and more.
To be simple and short, if it is merely a sport in your region, then go on and take part in their classes, particularly if you can find classes that are managed by Muslims although not really necessary AFAIK.
Hopefully other brothers will guide you better. I am not so familiar with the region that you live in. And if you need Fatwa you can ask a Mujtahid.