Missed your question there.
Have you checked the RBE debug log? Check /wp-content/bp-rbe-debug.log and see if any error messages are in there.
Also, when you attempt to fill in your GMail info on the settings page? Do you get any error messages after submission?
About the code, RBE doesn’t use fopen_url() or exec(), but it does use set_time_limit().
See if the VPS in question offers a free trial. If so, test RBE there and see if it works.
Your post gave me an idea to start a wiki page about webhosts that fulfill the requirements of RBE! Will try to gather some info from some of the RBE testers out there.