The other issue with using a lathe to power a belt sander is speed. Lower speeds as found on lathes will work, but with a separate motor and maybe a belt drive much higher speeds would be possible. Belt sanding works much better and much faster at higher speeds.
There's also the sand and sanding dust – probably better to have that away from your lathe if possible.
If you don't have any other space , OK, you have no choice, use the sander in the lathe. But if you do have room I would go with separate motor and probably a belt drive to allow step up and step down of speeds. JD
PS with any belt sanding – if you are sanding steel making sparks , after sanding some wood – clean all the wood dust out first, and keep an eye out for fire/smoke for a while after sanding. Don't ask me how I know this (it was just some smoldering wood dust, not a serious fire, but still worth some care)