It is not hard to do if you are (a) flexible or (b) lucky.

I have a return J LHR-SYD trip for two people booked for next week (I used my AA miles). I could have gone F but I didn't have quite enough points and didn't feel like buying them. In my case it was (b) - lucky because getting an award premium cabin flight just before Christmas is not easy, but at the time I booked there was both F and J.

F is often easier to find than J for the above reason.

The outbound is on BA and QF and the return is on QF only.

I don't use BAEC but from what I've read the award search engine is crap. I use AA or QF or ExpertFlyer to search. ExpertFlyer is limited usefulness as you can only search one airline at a time, but AA and QF will search everything. If you don't have a AA or QF account, then set one up and start searching. QF has better availability to higher tier (e.g. OW Emerald) flyers, so you just join QF you will be OW Zero for search purposes.