I've heard someone say that "一匹{いっぴき}" was used at the end of the original Godzilla movie, with Dr. Yamane saying that Godzilla wasn't the last one of "them".
Why did he use that, as opposed to "一頭{いっとう}"? Do Japanese speakers sometimes use the wrong counter to be ironic, or is there another reason?
There is certainly a reason for that. In this case, it is for expressing Yamane's (or the human kind's) derogatory feelings toward Godzilla.
The counter 「[頭]{とう}」 simply does not carry that derogatory connotation among us Japanese-speakers; It can only be neutral. In case this is what you are wondering about, the size of Gozilla does not matter as Godzilla is way too large to begin with. The general real-life rule of "匹 for smaller animals and 頭 for larger animals" does not apply here.
「[一匹]{いっぴき}」, however, does just the job when used in the right context. With it, you can "say" out loud "little f*****" without actually saying it.
Believe it or not, 匹 can even be used to refer to humans if they are your enemies. In sports, excited fans often use it to refer to the opponents. You will hear people say something like 「さあ, あと一匹だっ！」 at the bottom of the 9th inning with two outs in baseball.
Googling the two terms in Japanese, there are a great deal of discussions among the Japanese about when to use which as clearly ambiguous to them as much as it is to you. One of such quoted the definitions from Dictionary of How to Count (『数え方の辞典』) written by Asako Iida (飯田朝子).
【匹】
・大型ではない生物全般。
・小型の哺乳類。
・小型の爬虫類、両棲類。
・魚類。【頭】
・大型の哺乳類。
・大型の爬虫類。
・学術的な希少動物。
・学術分野でチョウ（英語の数え方の直訳）。
・人にとって重要、貴重、有益な小動物。（実験動物、カイコなど）
・人が訓練した役に立つ犬。（警察犬、救助犬、盲導犬など）
・まれに大型の鳥類。（ダチョウなど）【恐竜】
匹、頭、体小型のものは「匹」、大型のものは「頭」で数えます。骨格標本や模型などは「体」で数えます。
More or less the definition of "小型の爬虫類、両棲類" for 匹 seems suited for this case. One may say it is an expression of belittlement. But also it can be taken as an expression of endearment by the diminutive expression.
One of the discussions said NHK does not have clear definitions in use of the two terms and neither for average native Japanese speakers. Some says, 頭 is used for any animals large or small in zoology. Some suggested that there is some historical legacy in native speakers count in some unit the way they do (e.g. rabbits counted as 羽 so that Buddhists who was forbidden to eat 4 legged animals can eat it as birds). Some says 匹 is used for hunting, distancing hunters from respecting the games.
All in all, we can read in a lot and try to see meanings behind it. But I am not sure if the authorial intent can be clarified in a clear cut manner here. I, as an average native Japanese speaker, certainly can't.
This answer is supplementary, but using the counter 匹 for large animals is not so unnatural in daily conversations.
For example, Google returns roughly the same number of results for "2匹の象" and "2頭の象". I haven't checked out each of the results, but it's very unlikely that all those people using 2匹の象 are elephant haters.
If explicitly asked "what is the correct counter for 象 in Japanese?", many people will answer "頭".
But in practice, we often unconsciously use 匹 for all kinds of animals, including elephants, whales or dinosaurs. Basically, don't think using 匹 for huge animals is always ironic or derogatory.
