It's possible a problem with Rufus, not with elementary OS. You could download the elementary ISO yourself here: https://elementary.io/
After that you can point to the downloaded image from Rufus. You could also use another tool like YUMI, UnetBootin or Balena Etcher.
This solution worked for me so far:
Edit /etc/bluetooth/input.conf and under section [General], uncomment the line IdleTimeout=0
# Set idle timeout (in minutes) before the connection will
# be disconnect (defaults to 0 for no timeout)
Interesting that defaults to 0... but in my computer bluetooth was disconnected after roughly 10 minutes.