Is there a way of sending lots of large files online, or would it make more sense to have a stick for them and send them registered post?
A stick is very old school

And open to corruption.
Depending on the size of the files I'd be looking at Google Drive. Big, big, big files might mean charges for memory though that'll be the same with any commercial provider and if you're willing to pay registered post fees........
You could either share your files through that, or set up a new account for the two of you to work out of.
You'd probably need some kind of protocol to make sure you weren't working on the same file at the same time.
It would be less "sending" the file, more downloading and then re-uploading the file. Again, for big files upload/download speed will be important.
In any case, the commercial options all have "free" basic levels (as far as I know) so you can test what works best for ye easily and cheaply enough.