"From looking at the avios earning section of the BAEC sticky, it seems that if any of my journey legs are operated by Vueling, there is no way of earning avios for that leg in a BAEC account. Is that correct?"
Yes

"One option is to book a flight on ba.com from LHR-BIO. It is IB flight number (operated by Vueling). The same flight is more expensive on ba.com than Iberia.com, which in turn is more expensive than Vueling. Is there any advantage to booking on ba.com? "
No. Actually, if you want to select premium seats on the Vueling flight (first row, blocked off middle seat), you can only do so if you purchased the ticket on Vueling.com

"If booked in this way, can I access the BA lounge in LHR T3 with BAEC silver status?"
As a OW Sapphire holding an IB-issued ticket, I was told no. As the flight is operated by Vueling, this did not qualify. I argued that the ticket had an IB flight number and was purchased on the IB site specifically for this benefit and was told that "on this occasion" I would be permitted inside.

"I can't seem to combine this LHR-BIO flight in a multi-city on ba.com (even to go LHR-BIO/BCN-LHR) but I can book one-ways for these. This seems odd."
You could buy one ways on the Vueling site for flights operated by Vueling (at a reasonable cost.)

"In this scenario, if the final leg was IB operated by BA, I would definitely earn avios through BAEC and have lounge access in BCN, correct?"
Correct, but given your situation, I would consider a Priority Pass lounge.

"I may end up booking a series of one ways or (gasp) be forced by others in my party to include some Easyjet flights if I can't work out an advantageous way.
Easyjet does not fly to LHR, Vueling does. I like the blocked middle seat front row arrangement better than Easyjet's premium seat options.

BTW, if Avios are that important to you, you could buy your ticket on IB and have the Vueling segments credited to your IB+ account. Then transfer the miles to your BAEC. Check first the restrictions on merging miles for these who are not already members on both programmes.