Very interesting to think about this now: we'd capitalise the first word of the name if it was at the beginning of a sentence, or perhaps standing in a list or index. Red grouse.
The genus (or "noun") we could capitalise as a proper name in English, although not necessarily, as in red Grouse. The "red" is just an adjective and wouldn't normally be capitalised in English - unless the WHOLE, taken together, the noun plus the adjective, IS the proper name, in which case it would be correct (and perhaps desirable, for clarity), to capitalise both words as one "combination" proper name, Red Grouse, which is what I'd vote for.
But it may still not be correct to capitalise each word, such as in the putative case of Red-Backed Shrike, as the hyphen links "backed" to "red" as one word; it should therefore read Red-backed Shrike. Would the capitalisers be content with that, as it bends only to laws of English grammar? Please correct me if I'm wrong.
Of course the laws in German for capitalisation for nouns are different, and the French noun-adjective order is back-to-front compared to English, (or perhaps ours is front-to-back compared to theirs!) But still, in French, the question would be whether to capitalise both words as one whole combination proper name.
A quick flick through my collection proved quite eye-opening:
lower head-casers:
Gilbert White, Selborne 1789, Illustrated edition (2004)
RSR Fitter & RA Richardson, Collins Guide to British Birds, Revised (1966) reprinted 1973
James Fisher, Shell Bird Book, 1966
Christine Jackson, British Names of Birds, 1968
Montagu, Ornithological Dictionary, 1802, ed 1831
Macleod, Key to the Names of British Birds, 1954 (Jobling gives this as a reference and source in his Dict. Scientific Bird Names, 1991, but doesn't follow his example.)
Stephen Moss, Mrs Moreau's Warbler, How Bords Got Their Names, 2019.
Betwixt and betweeners:
Charles Swainson, Provincial Names and Folk Lore of British Birds, 1885 (uses Capital letter for the first name only)
DAS KAPITAL-IZERS:
Alfred Newton, Dictionary of Birds ,1896
RSPB Where To Discover Nature in Britain and Northern Ireland, 2009
Heinzel Fitter & Parslow, The Birds of Britain and Europe with North Africa and the Middle East, 3rd ed. 1974
William Lockwood, Oxford Dictionary of Bird Names, 1984
James Jobling, Dictionary of Scientific Bird Names, 1991
Francesca Greenoak, All the Birds of the Air, 1979; British Birds, Their Folklore, Names and Literature, 1997
John Braidwood, Ulster-Scots Bird Names, in Ulster Folk Life, 1965
Strangely the RSPB seem to have changed sides. Score currently 7-7!