I've got side-tracked a bit with differentials of exp on Lie groups and it looks like it's not too complicated to work out. For specific groups used in applications it's possible to just use symbolic computation on matrices like here: https://gist.github.com/mateuszbaran/9b0e65b23afc23146850595d27a913dc, which shows how to do it for SO(3). In principle we could do it for any matrix Lie group.
I've got side-tracked a bit with differentials of exp on Lie groups and it looks like it's not too complicated to work out. For specific groups used in applications it's possible to just use symbolic computation on matrices like here: https://gist.github.com/mateuszbaran/9b0e65b23afc23146850595d27a913dc, which shows how to do it for SO(3). In principle we could do it for any matrix Lie group.