The hyphenated and double-capitalized "Pre-Raphaelite" appears to be the spelling that is most common and most commonly recommended. The American Heritage Dictionary says "also pre-Raphaelite".
It's unclear to me whether there is any generally applicable rule that explains the capitalization of "Pre-" here. Sven Yargs's answer to "Hyphens in words with prefixes like neo-Nazism" says that the prefix neo-/Neo-, which I think seems similar, often is spelled with lowercase N in words like "neo-Darwinism, neo-Expressionism, neo-Freudian, neo-Gothic, neo-Malthusian, neo-Nazism".
Perhaps some kind of distinction could be drawn based on the fact that expressions like "Pre-Raphaelites" and "Pre-Raphaelism", in their most common usage, do not refer to "precursors to a movement of Raphaelism" but "followers of a movement inspired by the painters who preceded Raphael". In other words, it's not formed by prefixing "pre-" to "Raphaelism", but by combining "pre-" with "Raphael" and then affixing "-ism". Nevertheless, spellings with lowercase P are in fact attested in sources from various time periods; the OED provides the following quotations that illustrate this:
1850 Athenæum 27 Apr. 455 He [sc. W. H. Deverell]..has painted..heads and limbs with an appreciation of higher and better example than those which enthral the minds of his pre-Raphaelite brethren.
1974 J. Christian Pre-Raphaelites in Oxf. 27 Port Meadow has many pre-Raphaelite associations. Hunt would paint here... Only Ruskin said its beauty was spoilt for him by the sight of the nearby slums of Jericho.
1999 Marie Claire Health & Beauty Sept. 42/2 Hairdresser Marc Lopez swept loose curls into an unstructured up-do, then a headful of pre-Raphaelite waves.
Other possibly similarly constructed words that might be compared:
There is no general rule saying to capitalize the prefix pre- when a hyphen is used to combine it with a capitalized word
The Chicago Manual of Style, edition 16 says
premodern, preregistration, prewar, preempt, but pre-Columbian, Pre-Raphaelite (pre lowercase or capital as in dictionary; lowercase if term is not in dictionary)
There is no general rule saying to capitalize a word that refers to a cultural movement or art style
The Editorial Style Manual of The Pennsylvania State University suggests that "cultural movements and styles" should not be capitalized in general, but only when they are derived from proper nouns:
Cynicism, Doric, Gothic, Neoplatonism, Pre-Raphaelite, Romanesque; but baroque, classical, cubism, Dadaism, modernism, neoclassicism, postmodernism, romanticism.
Different preferences are attested in prescriptive sources
I found a book from 1911, "135000 Words Spelled and Pronounced (Preferred Form)", by John Hendricks Bechtel, that favors the spelling "pre-raphaelite", which is now uncommon:
Proper nouns and adjectives preceded by pre- are written with lower-case and a hyphen: pre-adamite, pre-mosaic, pre-raphaelite.