No, you do not want to stream from the internet. You want to send the STL file to the Pi. Slice on the Pi, and during slicing, stream the GCode straight from the slicer on the PI to the printer. That's what I'm suggesting. This means it does not really matter if the slicing takes 10 minutes, as you can start printing as soon as you have the first layer buffered.
(And with some tweaks in the CuraEngine you could have the first layer in less then a second)
GCode files are usually a factor bigger then STL files. And if you convert the STL to CTM, you would have even less data to transfer. Cutting down on possible failures. Especially after seeing GCode files well over a few 100MB.