If such a thing happened, it would most likely have been during the approach and rendezvous of the LM ascent stage with the CM, as Conrad and Bean were returning from the lunar surface. The most challenging part of the mission would be behind them and there would be a little window of time where they could maneuver around on the RCS thrusters without any risk to the mission.
I skimmed the annotated "Flight Journal" transcript for this period and didn't find any sign of Bean flying the spacecraft. About the closest thing I saw was Pete Conrad suggesting tongue-in-cheek that Bean was so eagerly maintaining the guidance solution in the AGS backup computer that they could perform the CDH burn (one segment of the rendezvous approach) on it instead of on the primary computer; someone could interpret this as "Bean flying the spacecraft", although (a) it would really be a computer flying the spacecraft and (b) Bean more often than not would be operating the primary computer as well as the AGS.
Looking at the LM cockpit voice recorder transcript, though, I did find this rather interesting segment which occurred in the "long pause" from 142:22:06 to 142:22:55. LMP = Lunar Module Pilot (Bean); CDR = Commander (Conrad).
05:22:22:05 LMP I'm ready when you are, babe.
05:22:22:08 CDR Okay. There's a - an ENTER. There's a PRO, 54.
05:22:22:16 LMP Okay.
05:22:22:20 CDR What do you want?
05:22:22:21 LMP I'm letting it drift just a minute. Okay. Pitch up and left.
05:22:22:27 CDR That's enough. That's enough. Now that's one pulse; that's it.
05:22:22:31 LMP I know it. Down and right. Up and left.
05:22:22:55 LMP That did it.
05:22:22:56 CDR Okay. There you go. Beautiful, Al.
05:22:23:03 LMP Stick with me, babe.
Now, I am not really certain what's going on here. "Enter... PRO 54" is Conrad operating the guidance computer DSKY, which is typically Bean's job. Bean's "I'm letting it drift" suggests he could be at the commander's controls. His "pitch up and left", "down and right" could be instructions to the commander, but they could also be announcements of his intent on the controls. Conrad's "that's enough", "that's one pulse", "there you go, beautiful, Al" could be instructive hints and reassurances.
If this is in fact Conrad letting Bean drive, it seems to have been just less than one minute of maneuvering. Perhaps Bean just got to test out the rotation controls ("pitch up and left") and then RCS translations ("down and right... up and left") so he could get a sense of the feel of the LM. Bean's "Stick with me, babe" could be a tongue-in-cheek take on fighter-jock swagger -- ironically equating wiggling the LM controls for a few seconds with being a hotshot pilot others should look up to.
The episode "That's All There Is" of the HBO series "From The Earth To The Moon" depicts Bean getting to fly the LM after the ascent, while the LM was behind the moon and out of contact -- which would be sometime between 142:30 and 143:16, which doesn't match the timing of my excerpt; none of the LM onboard transcript in that period suggests to me that Bean was flying the spacecraft, and there aren't any significant gaps in the transcript.
The HBO series was based on Andrew Chaikin's book A Man On The Moon which gives essentially the same account:
...during the rendezvous, [Bean had] been slaving away with the backup computer and the navigation charts while Conrad flew the lunar module. They had one more burn to do, and then they would have it made. And Conrad had said to him, "Why don't you just quit after this midcourse, and relax and enjoy it? You can take a minute and fly this vehicle." Startled by Pete's audacity, Bean wondered, wouldn't it put them off course? No, Conrad assured him, whatever digressions they made would be easy to correct. Bean was reluctant -- surely mission control would know. Conrad laughed, "Not on the back side of the moon, they won't." Bean realized Conrad had planned this perfectly. And for a few minutes Bean had his hand at the crisp, responsive ascent stage. It was a moment that Bean would always remember as pure Pete Conrad, that in a small craft somewhere over the far side of the moon, he had taken the time to share with Bean a flying experience that even most astronauts would never know.
Chaikin's book was based on personal interviews with the astronauts; Bean and Conrad were his first subjects, interviewed in 1985, 16 years after the fact.
I'm really not sure what to conclude from this. Bean said it happened during the backside pass, but it's possible that he misremembered the timing after 16 years. I'm reasonably sure that it happened, at least.
How did Conrad let Bean drive? Did they switch seats?
No seats in the LM; they would have just had to scoot past each other. "From The Earth To The Moon" doesn't show them switching positions, an odd oversight, but there's only one set of hand controllers on the commander's side of the cabin.
How severe of a "wasn't supposed to" was this? Was it unscheduled but otherwise okay, or did it break some protocol? Did anyone get reprimanded?
I'm sure it would not have been a big deal. I don't know of any mission rules explicitly saying the LMP wasn't allowed to fly the ship. There was almost nothing the LMP could do at the commander's controls that would cause a serious problem at that point in the mission. Even flying the guidance platform into gimbal lock would be only an inconvenience. Letting a 14-year-old drive the family car for a lap around an empty parking lot would be far riskier -- there are no lamp-posts to hit in low lunar orbit.
I didn't see a mention of anything about letting Bean fly in the ascent/rendezvous portion of the crew debriefing; even if there was no particular rule against doing it, they might want to err on the side of keeping it quiet for a few years.