Hmm, the screenshot does not necessarily look like wrong weights, it looks also as if some polygons could be wrong sorted or have a wrong normal. Can you check if there is a submesh with a different texture or material, or a different smoothing group?

When the weights were wrong assigned by the engine, I'd expect that it would happen with other models too. So my first guess is that's something related to that particular model. But you can check:

First, export the whole animation to FBX vertex frames. If the problem then still happens, it's unrelated to bones weights.

Otherwise, export to FBX bones but import as vertex animation. If the problem then still happens, the FBX file already contains wrong weights. It might be a problem of the FBX exporter then.

When the problem disappears when importing as vertex animation, it could indeed be an engine bug. In that case, can you send us the FBX model?