BA's baggage allowance for Oneworld Emerald in Premium Economy is 32 kg x 3, and this is extended to other passengers on the same booking as yours (note, only those on the same booking as you will receive your extra allowance).
You personally would not be allowed 6 pieces of baggage - it would be 3 pieces per passenger on your booking, by the way, in case BA agent was not clear.
Baggage allowance is determined by the operating carrier of the most significant sector and since the most significant sector would be JL on the way back, I'd have thought that JL's allowance would be applicable.
Extra weight allowance for Emerald (32 kg instead of 23 kg) is very much a BA thing, and I believe the baggage allowance on JL would be 23 kg x 3 (23 kg x 2 for PE, plus 23 kg x 1 for Emerald), and the extra piece of luggage would not be applicable to the extra passenger without status, so I believe JL agent would be correct.
JMB status holders are treated differently from non-JL OW status holders so it is best not to rely on information for JMB status holders (same applies to other OW carriers' own members vs others).
Regardless of baggage allowances though, it's worth remembering that carrying 3 x 32 kg luggage + cabin baggage is not easy even on a trolley and you might like to think about the practicality (I do it very often and it is never easy, and 6 pieces of full size luggage would never fit in a standard taxi either).