They are not awarded a new series, so that's not the correct rule reference.
Yes, they are.
5.1.3 -- When a scrimmage down ends with the ball in the field of play or out of bounds between the goal lines, a new series is awarded to:
(e). R, if K legally kicks during any scrimmage down and the ball is recovered by R, is in joint possession of opponents, goes out of bounds or becomes dead with no player in possession.
Edit -- I need to add more to this because I think there's some nuance that's getting overblown here. I get that in the scenario, there is a foul that is causing the replay of fourth down -- therefore, R is NOT getting the ball, therefore they clearly aren't getting a new series awarded. However, there's no explicit rule coverage or case book play (that I've found) that covers this exact scenario and I think it would be helpful to think of it from a philosophical point.
The argument is that the clock would only have been stopped at the end of the down *because* of awarding the new series, and since the new series is now off the table, the clock would not have otherwise been stopped. The clock stoppage due to the foul supersedes the new series stoppage, and therefore the new series stoppage is irrelevant since the new series wasn't awarded due to the foul. If this is true, then it should be expanded to all situations -- if the clock was stopped after an incomplete pass, the foul supersedes the incompletion for clock stoppage. With the penalty applied, the incomplete pass is no longer applied as the down is replayed, so the clock stoppage due to the incompletion is irrelevant, right? Of course not.
The problem lies that there are multiple reasons why the clock was stopped at the end of the down. After the scrimmage kick, the clock is stopped immediately when the ball becomes dead because it is apparent that R will be awarded a new series. You do not let the clock continue to run for a few seconds after the ball becomes dead to determine if other action during the down prevents R from being awarded the series. The BJ kills the clock immediately because the ball is dead after a scrimmage kick which would nominally award a new series to R.
Here's my philosophical approach -- if a down is replayed due to penalty, the clock will start on the ready or snap in the same manner as it would *had the penalty been declined*.
If you decline the penalty and don't replay 4th down, the clock starts on the snap. Therefore, if you replay 4th down due to penalty, the clock also starts on the snap.