Does a soldier remain flagged for ABCP if they PCS'd and his last unit did not upload the supporting documents?

There are no supporting documents to upload and nowhere to upload them to. Those documents you're referring to are only used to process an ABCP separation. As you said, your unit's program is all messed up and so are many other unit's programs. The person can be flagged and not have the correct documentation, but they can't be separated without the correct documentation, if you understand what I'm saying. As for uploading, there is no place to upload the documents for a current flag. Emilpo only creates the flag, it doesn't store data. OMPF/Iperms won't accept those documents unless it's part of an approved separation file. No JAG at the gaining unit command is going to accept a separation without the supporting documents from the previous command, and those documents being in correct accordance with the regulations.
So, while the SM will remain flagged, it would be almost impossible for the next unit to use that as a grounds for separation.

The easiest way to fix this is have the Soldier get taped. If they are compliant submit a 268 to have the flag removed, since PCSing they are probably not already enrolled in the unit's ABCP and thus will not be hurt by not making progress if they are not compliant.

According to regulation concerning FLAG's, an overweight FLAG is transferable. The SM is also required to take an APFT/Height and weight within 90 days of reporting to their new unit. usually, the Battalion Aid Station has qualified personnel to administer the "Tape test", and the Company should also have an NCO qualified as well (Additional duty). You can also attempt to contact his last unit, and ask them to forward the documentation. I would speak to the 1SG regarding this discrepancy in your ABCP program, as I am fairly certain that your Battalion level tracks PT and Height and weight failures (It is a reportable item to higher during Command and Staff meetings), and getting this program correct is a very important thing.

Another aspect you have brought up is giving your SM a "Fair shot". If this SM had a 268 submitted for being overweight, and then PCS'ed, you could ask the 1SG to inquire with S-1 as to when the 268 was submitted, It is unfortunate, but not unheard of, for a flag to not be lifted by accident. I assume it doesn't need to be stated that if it seems the SM is borderline, that you counsel the SM in regards to this.