I'm shooting almost daily with 1D4 & 5D3. They're a pretty good matched pair with different strengths. My go-to body for any type of action shooting is the 1D4, it's much better than the 5D3 in this regard. Give both bodies their best chance of delivering their highest buffer clearing performance with the fastest cards you can afford. They make an instantly tangible difference. I'm also a great fan of the x1.3 crop APS-H sensor found in the Mk4. Depending what your client wants to do with the shots, pushing the 1D4 to 6400-8000 iso is very do-able provided your exposures are perfect. A modest dose of NR in Lightroom will be all they need for good, usable output.
I love both cameras, they're both first class pieces of kit either of which will deliver the goods in most situations. For more extreme conditions or output requirements, each camera will have it's day. On the level of handling and ergonomics of any 1-Series body will leave the rest gasping for respectability. A minority of shooters will challenge this position, but 1-Series users don't tend to use these superior bodies to look cool, they just know from experience that they just can't be beaten. The loudest 1-Series critics probably haven't used them in prolonged demanding conditions, or maybe not at all.
Get the 1D4! $3k should get you a beauty.