UVFITS is a file format that has a history of use in AIPS, so it likely won't be much use here.
It looks like an easy way to get what you're looking for is to use the export_uvtable task of uvplot (see documentation here). Since uvplot is not a standard CASA library, you'll have to install it with pip from within CASA (if you are using a mac and CASA 5 or above):
pip install uvplot
Then run split to isolate the visibilities you wish to extract, e.g.:
split(vis='FULL.ms', field="0", spw="0~2", outputvis="TEST.ms", datacolumn="data")
And finally run export_uvtable
from uvplot import export_uvtable
export_uvtable("uvtable.txt", tb, vis="TEST.ms", datacolumn="DATA")
The resulting uvtable.txt file contains columns of u, v, Real(V), Im(V), and weight.