Answer

Regarding a tattoo that is not writing but a drawing, the halakhic decisors disagree. Maimonides implies that any tattoo is included in the prohibition, but in Pitchei Teshuvah, Yoreh De'ah 180:1, opinions are brought that prohibited only writing. Even so, it is still difficult to be lenient about this.
As far as I know, there is no difference between a tattoo for the sake of idolatry and an ordinary tattoo. True, Maimonides brings this in the laws of idolatry (chapter 12), because in his view the source of the prohibition is that this was the practice of idol worshippers. But not everyone agrees with him on this. Some see this prohibition as one of injuring the body, and the like. But even according to Maimonides, I have not seen that he interprets the reason for the verse in a way that would prohibit only a tattoo made for idolatry. That is only the rationale of the law, not the definition of the law itself. Therefore, in my opinion, one should not be lenient about this.
To sum up, in my opinion it is difficult to be lenient regarding a tattoo that is not writing, and certainly regarding a tattoo that is not for idolatry. And in general, it seems to me that it is simply not appropriate to do this.