Python Mosaicing S1 SAR products problem(s)

Maybe your input products use a compression? 1.7 GB for a mosaic does not seem to big to me.
But the difference between SNAP Desktop and snappy is strange. I think somehow the configuration must be different.
As far as I can see the operator uses the data type of the source bands for the target bands. This should not make the difference.
Why the GeoTIFF-BigTIFF hangs after a while might be caused by your memory settings.
Please have a look at the following thread.

You can use NetCDF4-CF as output format. This uses compression by default and does not have the 4294967292 limit.
For GeoTIFF-BigTIFF you can configure the compression:

You can add those properties to the etc/snap.properties file in the installation directory.