Not being particularly well-versed in how well (or not) shipyards are working, I can only go on the cost of pirate caps, which seems … rather high. From what I’ve heard of complaints about them, it seems that the/a problem is that the shipyard only drops a few crazy valuable BPCs when they explode. This concentrates wealth, disincentivizing the large fleets that are needed to bring them down in the first place.
My proposed solution: change the faction cap BPCs to require special faction cap components (armor plates, drone bay, shield emitter, jump drive, etc. etc.), the BPCs for which also drop in shipyards (or shipyard-like) sites. Make the hull BPC drop in greater numbers to drop their value, so that more of the income from the drop comes from the component BPCs. Make the component BPCs drop at multiple parts around the site, or even out in other, lesser sites. I suppose something like how Incursions work: there are lots of smaller sites that gradually build up to the big assault that drops the final hull BPC.
This way, smaller fleets can tackle the smaller sites and still score BPCs that bring decent income, and the shipyard itself doesn’t turn into some giant all-or-nothing pinata.