What would be the British English equivalent to "The Elements of Style"?

There is Fowler's Modern English Usage for British English. It is much longer than The Elements of Style, however.

Though I have never used it personally, you may want to check out The Complete Plain Words. That appears to be a much more concise, pamphlet-style guide to British English.

As far as your second question goes, I don't know of anybody (either in the US or UK) that thinks it is better to make your writing more difficult for readers. I probably cite this essay once a week on EL&U, but Orwell's Politics and the English Language is a small 'British' style guide in itself. Orwell puts forth several stylistic rules, or rather things to avoid in order to make ones writing more clear. I only bring it up because Orwell is British and he puts forth a great argument for clarity and simplicity in writing.

Fowler too advocated for clear writing (see the section on 'Approach'):

Henry W. Fowler’s general approach to English usage was to encourage a direct, vigorous writing style, and to oppose all artificiality — firmly advising against unnecessarily convoluted sentence construction and the use of foreign words and phrases and archaisms. He opposed all pedantry, and notably ridiculed artificial grammar rules not warranted by natural English usage...

I don't know of one, but from a quick scan, I think most of the guidance stands equally well in the UK...

Writing should be about communication not obfuscation, so following a few simple rules [Perhaps simple is the wrong word to use in relation to the English language, but...] and use of a good dictionary should be enough.

I can highly recommend:

"Eats, Shoots & Leaves: The Zero Tolerance Approach to Punctuation"

A book by Lynne Truss

[Available at all good book shops]