A couple of things I’d consider:
* Pay for Strava for a month, and use their heat map to plan routes. They’ll show you what routes the local cyclists use, and given none of the Spanish I know are particularly suicidal, they’ll be avoiding the worst roads.
* As much as possible avoid any road “N-xxxx”. These are national roads and generally have more traffic. The quieter local roads have the Community or Province code at the start (NA-678, BI-631, M-510, EX-202, V-123, etc). There’s no way you’ll be able to avoid them completely, and it’s not instant suicide to stray onto one, but generally speaking if you have an alternative I’d probably use it.
* I’ve not done the route, but from a quick look at the map I’d probably do Bilbao – Vitoria – Logroño – Soria – Teruel – Requena – Alicante, and avoid Valencia. The few times I’ve driven round Valencia it’s been pretty horrible (although there might be a bike route that I don’t know about), and the coastal road down to Benidorm will definitely have a lot more traffic than further inland. (I’ve got a place just south of Alicante, and I avoid the N-332 as much as possible on my bike – it’s the same main road that runs past Benidorm up to Valencia…)