ASTAP does allow saving as a 16 bit FITS file:
File -> Save as FITS file
On the Save file as window that appears select the down arrow on the Save as type field. One of the options is 16 bit FITS files (*.fit*).
Select this and change the name of the file as required and save.
Theres some fun for anyone using Python, the Astropy library discourages the writing of 16 bit FITS files and I think it defaults to 32bit.
I have some memory of being able to convert 32 bit files to 16 bit in Python by using numpy (get the data into a numpy array and convert using np.int16(data) and then write the data to a file) but I could be mistaken.