闘魂 (toukon) is made of fight+soul, and seems likely to be used as a literal translation of 'fighting spirit' or in colorful descriptions of a burning urge to do battle, judging by the dictionary. I've never seen it in actual use.
闘志 (toushi) is made of fight+will, and is more likely to be used for 'has fight in him' and to indicate readiness to take on an opponent (possibly but not necessarily in a physical fight). It can also, I believe, be used for anything 闘魂 can be used for. I've come across it a number of times.
I've seen 士気 (shiki) even more, though, which is more like 'morale (of the troops)', but with a little rephrasing can be used to express much the same sentiments as 闘志 in many cases. I feel like 士気 is used more often for groups and 闘志 more often for individuals. I haven't encountered them in actual reading enough to be certain of that, but the dictionary examples strongly suggest it and my reading experiences do match that.