What holds more weight, a soldier not overcoming a bar on his second review or a potential med broad process?

Clean your ranks. Not just pme failures, but also anyone who hasn't taken an APFT in a long time. Scrub your books and check you may have quite a few. If they cannot take an APFT they cannot go to a pme and you can bar them for this. You'll be amazed though how many Soldiers that get barred end up in the ides. It is a lot of work though and very frustrating. Or do nothing and the same Soldiers will end up in the ides later once they get a Pcs they don't want. I have seen it so many times. Go get it or wait for it, either way you'll save very few by enforcing something as simple as them taking an APFT. The docs should catch it at the one year mark of temporary profiles, but not so much.

MEBs are initiated at the direction of the soldier's PCM (typically a BN PA or Surgeon) as outlined in AR 40-501 Chapter 3. The soldier typically does not have the choice to trigger his/her own MEB (I say "typically" because sometimes soldiers really want to stay in and try to perform on a P2 profile). There is a process to determine when a soldier is eligible for medical separation based on the condition/diagnosis and when they meet MRDP. It is not uncommon to have soldiers threatened with administrative separation attempt to seek medical separation for a condition that does not meet the criteria in AR 40-501, or, has not met the MRDP, yet.
During an administrative chapter, a separation physical is conducted. If there is a condition that meets the medical separation standards (very rare as they typically have not had the work up to determine MRDP) then, both the MEB and chapter process proceed. Once both are complete, the 1st General Officer in the CoC determines the disposition (in my experience, they have always gone with the chapter).
Pertaining to the APFT, per new medical readiness guidelines established on 01 June 2016, there is no longer a "recovery period" following the termination of the temporary profile, so, the soldier can be expected to perform an APFT the day after a profile expires. If they need additional recovery, then, they need a new profile. Army Directive 2016-07 outlines the new readiness guidelines.
Not sure what kind of unit you are in; if you have your own BN PA, I would suggest you and your 1SG engage him/her for guidance. I and all of my fellow PAs are engaged with our 1SGs to help eleviate some of these issues.

MEB trumps a discharge packet. I am dealing with this now and the SJA told me that. So I still continued to do in my case an UNSAT packet so that if he is found fit for duty I have the packet already to submit and give him the boot. So yes continue to do the APFT packet so that you can be ready when you get the MEB findings.