Where did the phrase "batsh*t crazy" come from?

I started to type up an answer summarizing several theories about its origin and first use, but then I found that since the last time I tried to look it up, the Oxford English Dictionary has in fact added an entry for this term (in the online edition), along with some helpful citations. Turns out the phrase didn't originate with Hunter S. Thompson, or with Kubrick's Col. "Bat" Guano in Dr. Strangelove.

Here are the definitions provided by the OED, along with the earliest citation for each:

1. A worthless or contemptible thing; rubbish, nonsense. Cf. bullshit n. (1950 M. Shedd Return to Beach ii. 156: "I felt the minute hand of that bat shit of a Judas clock stand up to me.")
2. Austral. Used in similative phrases as the type of something dull or uninteresting. Chiefly in boring as batshit. (1964 G. H. Johnston My Brother Jack iv. 58: "He would describe somebody as being ‘as silly as a two-bob watch’ or ‘dreary as bat-shit’.")
3. Crazy, mad, insane. Cf. bats at bat *n*. 1 b. Orig. and freq. in to go batshit (cf. to go ape-shit at ape n. Additions). (1971 W. Calley Lieutenant Calley 104: "Most of America's males were in Korea or World War II or I. They killed, and they aren't all going batshit.")
4. As an intensifier, esp. in batshit crazy. (1993 Toronto Life Aug. 6/4: "His mug is emblazoned with the words: full-blown bat shit crazy.")

So, while batshit crazy certainly does seem to be influenced by the expression bats in the belfry as you suggest, its first meaning, in use by 1950, was simply a variant of bullshit. This use continued and overlapped with the "crazy" meaning: further citations are given for definition #1 from Dean Koontz's 1985 novel Door to December and from Seattle Weekly in 2002. Also, batshit as a standalone word meaning "crazy" appears to be older than the two-word phrase batshit crazy, at least as far as the written record shows.

There's anecdotal evidence scattered around the internet, like in this Straight Dope Message Board discussion, that definition #1 was in common use in the US military during the 1950s. Someone else points out there that Hunter S. Thompson may have picked up the term in the Air Force, from which he was discharged in 1958. Apparently he used the terms batshit, batshit crazy, and/or batshit insane in Fear and Loathing in Las Vegas and/or Fear and Loathing on the Campaign Trail '72, but I don't have either book handy to check, and none of the claims I've found includes a full quotation.

From an inexplicably deleted answer is a 1983 cartoon by P. S. Mueller (the voice of Onion News Radio) captioned:

Full blown batshit crazy and still holding down a productive job

Mueller adds:

Since I first published this cartoon around 25 years ago, I have received more correspondence about it than anything I have done before or since. I guess it struck a chord. As far as I knew at the time or know now, the phrase "Full Blown Batshit Crazy" just sort of fell out of my head and appears to have entered some part of the language. I seem to have the dimmest memory of a guy I knew in my early teens uttering the word "Batshit" on the school bus and nearly killing me with it. I could have it all tangled up with another guy or made it up myself, but it certainly sounds like something Birdy would have said.

I'm not sure when Mueller was born, but he "began contributing drawings to the Daily Egyptian at Southern Illinois University-Cabondale" from 1967, so his early teens batshit memory was probably around the early 1960s.

The word crazy is a later addition. Scanning Google Books I find a handful of references starting from the mid-60s where batshit is clearly just a variation on bullshit (nonsense, rubbish) - which meaning still turns up even in 2001, but it's relatively uncommon now.

Here's a relatively early one from 1967 where the meaning is crazy. A decade later most references have this meaning, but the earliest actual "batshit crazy" I can find is this one from 1989.

I'm not sure it makes a lot of sense to call batshit a "euphemism" for bullshit, but it does seem to me that's how the word got started. In ordinary speech there's no meaningful difference between "That's bullshit!" and "That's crazy!". Bats in the belfry/batty had been around since 1901/1903; obviously people just latched on to that crazy connotation, thus popularising the word-pair.

This might be a stretch, but what the author says here makes some sense.

However, it was interesting to note the first recorded usage of each of the elements of the phrase: bat appeared as bakke in C.E. 1340; shit appeared as schite in C.E. 1308; and crazy appeared as crasie in C.E. 1583. Knowing as we do that a) modern English words beginning in sh or sch began life as Scandinavian words beginning in sk, and b) bakke, with its hard b and double k, is obviously also of Norse derivation, it wouldn’t be too improbable to guess that Vikings were running around Northern Europe describing last night’s pillaging as totally “bakke-skite.”

The term has on occasion been attributed to Hunter S. Thompson, who was a lover of bats and quite crazy in his own right, but I believe the term originated well before Thompson's time.