What is an English word which means 'bêtement'?

"Bêtement" is by no means associated with ideas of insanity, and even less of bestiality.

It is a very mild word to say "foolishly".

"Zut, j'ai bêtement oublié mes clefs !" = "Oops, I have foolishly forgotten my keys !"

and "tout bêtement" means "quite simply"

"Alors, je suis rentré tout bêtement par la fenêtre". = "Then, I quite simply went in through the window".

and "une bêtise" = a not really important blunder.

A Franco-British

The most common animal metaphor relevant here is that of cattle - so you can say in a bovine manner thus denoting their dull and stolid mechanical nature.

Bovine is apt in this figurative sense. Continuing the animal metaphors you may prefer mutton-headed, which basically carries the same meaning of being dull and slow-witted.

A good non-animal robotic metaphor is automaton-like.

The standard translations of bêtement are words like stupidly, idiotically, insanely or thoughtlessly, carrying the metaphorical meaning of "like a beast".

'Mechanically' comes up as the best fit. It is unusual to find a French word for which there is no exact equivalent in English, since English has a larger vocabulary than French. I can think of dozens of English words for which it is difficult to find a French equivalent, the most obvious ones being 'home', 'fair', 'cosy' etc. I'm not an expert, but it is sometimes said that whilst the English rely on vocabulary to shade their meanings and alter the nuance, the French achieve a similar effect through inflections of grammar, in particular through the use of the subjunctive.

Since writing this @ Jim has suggested the word 'mindlessly' for this particular meaning of the word betement. This would seem to me to fit precisely, though I do of course recognise that betement has other meanings.