What can I do to make my opponent play faster at Regular REL?
I'm surprised that you seem to think there is no possibility of official intervention at Regular REL.  From a recently updated official document on Judging at Regular REL:


  
  General Unwanted Behaviours -
  
  
  There will sometimes be issues that do not have official fixes, but
  need to be discouraged. These include, but are not limited to;
  
  
  Players taking unreasonable amounts of time side-boarding or making
  play decisions  
  Inadequate shuffling after a search 
  Asking for, or
  providing, strategy advice during a tournament match or booster draft
  
  
  In all of these cases, educate the player on better behaviour – for
  example, alternate shuffling techniques or the importance of allowing
  players to make their own decisions. Players continuing to exhibit
  specific unwanted behaviour after being instructed otherwise should be
  issued with a Game Loss.


A Judge should be available to advise persistent time-wasters that, actually, there is a limit to how long they can spend "thinking about their moves" in a game of Magic.  Obviously the Judge will be gentle at first, as this is only Regular REL; but really, no one is being done any favours , if new players are not told early on that they need to play the game at  reasonable pace, that is to say, fast enough that a full three games can be comfortably completed in the time allowed.

Don't be a jerk about calling a Judge over, but equally, you shouldn't be afraid of doing so. It's not being rude to ask, in a neutral and polite way, for the basic ground rules of tournament Magic to be enforced.