We eventually found an ATM (Bankomat) and withdrew 60,000 Forint (a little less than $300). Helpfully the machine gave us six 10k notes so we went to a little store and broke one of them, watching out for change in old Forint (worthless - and a fun Hungarian game they play with foreigners - Chris).
Then we went to what appeared to be the taxi line, carefully picked a taxi that was well marked and had a rate card, and enquired if he could take us. He said no problem, then promptly tried to switch us to a different car with a decidedly disreputable looking driver (by which Rod means he was built like a professional wrestler/vodka drinker and wearing a torn denim vest - Chris). We walked to a third driver, despite his many protests it was "just a colleague". At this point it miraculously became possible for the original guy to drive us. Probably we should have aborted completely based on the inauspicious start but we got in anyway.
The car set off, he started the meter, but it seemed to be going up pretty fast. Rod looked at the rate card and Lo! we were being billed at the Country Nightime rate, which takes 1,500 Forint per KM. Upon having this pointed out the driver got grouchy, suggested we get out, offered a flat rate of 5,000 Forint, and finally irritably said he'd take us to "official taxi". This seemed to imply he wasn't one. (I remain under the impression that he was arguing that he was an official taxi as people generally don't try to convince you they're running a scam - Chris)
The official taxi was driven by a friendly minimal-English chap. The rate was something like 300 Forint per KM, and actually seemed to correspond to city daytime. There was no country rate on his card at all, and even city night time - the most expensive rate - was only about 350 Forint per KM.
So, we escaped being scammed, at least once.