Why was the Bletchley bombe so called?

This has always puzzled me.

It looked nothing like the dessert.

If they thought it was in some way bomb-like, why the extra 'e'?

There are two different machines here, the British bombe and the Polish bomba.

According to the caption of this photo of Bletchley Bombe Wiring:

The name Bombe arises from the relentless ticking sound that the machine made, leading operators to remark that it sounded like a time bomb waiting to go off.

Jack Copeland's The Essential Turing:

In Chapter 5, Mahon says that the British bombe 'was so called because of the infernal ticking noise it made, supposedly similar to that made by an infernal machine regulated bt a clock' (p. 291). This story was well entrenched among Bletchleyites. The need-to-know principle meant that few were aware of the Polish bomba.

...

Why the Poles chose the name 'bomba' seems not to have been recorded. Rejewski's only comment was that the name was used 'for lack of a better idea'. As well as meaning 'bomb', 'bomba' is the Polish word for a type of ice-cream dessert -- bombe in French. Tadeusz Lisicki, who corresponded with Rejewski during the years before the latter's death in 1980, is quoted as saying: 'The name "bomba" was given by Różycki... [T]here was in Warsaw [an] ice-cream called [a] bomba... [T]he idea [for] the machine came while they were eating it.'

It goes on to give another story from declassified American documents, that the machine must have a stopping mechanism where part of the machine woud drop onto the machine and make a loud noise. But this explanation seems unlikely.

It's important to realize that bombe, the decryption device designed to decipher Enigma-encrypted signals, was designed by a Polish cryptologist Marian Rejewski in 1938. The device was named by him, bomba kryptologiczna, while an alternative to honor the creator was also coined: bomba Rejewskiego.

As you properly noted, it isn't a bomb. Probably the author gave it this name because it resembled a bomb. We can only guess and that won't really do anything.

In 1939, a year later, a similar device inspired by "bomba Rejewskiego" was produced at Bletchley Park by Alan Turing and later improved in 1940 by Gordon Welchman. During this time, the people obviously realized the same dilemma you're having about the device not being a bomb, but to keep the naming as consistent as possible, they named it a bombe - both to refer to the original name by Rejewski and to diverge from the "bomb" implication.

I think this is as close as you're going to get, unless you can get your hands on some historic documents containing the exact process of Rejewski's work, which I don't think you'll be able to get on the internet, but it's still important to realize that the Polish name was the initial one, so that's what you must be coming from.

See https://en.wikipedia.org/wiki/Bombe