You should be able to open the FBX file in Max or Maya, and the bones should work and the figure should look as it did when you built it.
If it doesn’t,.. then the most likely scenario is that you’ve done some thing wrong in the building process.
Max uses Biped (figure skeleton), and Physique, (weight mapping envelopes) which can be tricky to set up and time consuming to get right,.
If I were you,. I’d look at what distortions you’re getting in the model when you export/import into another program,.
then open your original model in carrara and check those areas to see if there are any obvious issues,.
perhaps even “Detach the skeleton” and have a look at the base model,. and the skeleton as separate items to see if there’s anything that could cause any issues., ....different bone scaling, are the bones named correctly,. are there any unconnected meshes, etc
As far as “what program should you build the bones in” .. that depends on what program you want to work with.
but adding bones can be done in several programs. many 3D programs bone systems don’t transfer easily or well to other programs.
perhaps another question should be ,.. Why,. are you trying to export to max / maya ?
Hope some of that helps