Firmo: "is there a best place online to look for cars in Spain (or Spanish-registered cars) similar to autotrader and gumtree?"
I don't know about a best place, but you could trawl through the likes of autoscout24.es or milanuncios, but you'll probably be shocked at the price of second hand cars with very high mileage and quickly lose the will to live.
As Tadd says, there are some expat dealers who may take a UK reg car in PX, but will probably insult you with their offer. I think there are quite a lot operating on the Costa Blanca, not so many down here on the CDS. The most well known are probably Robertsons & Bill Brady (not personal recommendations, I have never dealt with either and have no connection to either). A quick Google will find them.
"would anyone advise against bringing our UK car (right hand drive) to Spain and registering it? I appreciate that a RHD car isn't ideal in the long term, but our car is extremely ecomincal and reliable, and would make a great runaround or short term vehicle."
Again, as Tadd says, re-registering a car onto Spanish plates can be costly and a lot of hassle (e.g. changing the headlamps just for starters), and as you say yourself, for the longer term a RHD drive car is far from ideal.
Another option could be buying a Spanish plated LHD car in the UK, to drive down with what's left of your possessions after shiply have taken the bulk. There are a few long standing LHD dealers in the UK - unfortunately for you, most are down south (Google Basingstoke & Chesham LHD). And as I said previously, you will need your NIE in order to register a car in your name. See this thread about getting your NIE. I am sending you a private message with a further suggestion.
Hope this helps.