Should "japanese" be capitalised when used as an adjective

Solution 1:

Generally speaking, nations and nationalities are capitalized. This is always the case with things, like cuisine or history, that are closely associated with the the country. Thus Japanese cuisine (not japanese cuisine) and Chinese dynastic history (not chinese dynastic history).

There are a small number of exceptions, when the item described has a more remote connection. Thus

We'll use the good china

(not "the good China") to describe porcelain or vitreous dinnerware. And

japanned furniture

(not Japanned furniture) to describe black lacquered furniture. Opinion is divided on whether to capitalize english to describe the spin of a ball.

Solution 2:

In English country names, and their associated adjectives and languages are always capitalised.

In some languages, such as French, only the country name is capitalised - the language and people are not.

Solution 3:

As per the capitalization rules set out in GrammarBook.com

Rule 2. Capitalize proper nouns—and adjectives derived from proper nouns.

Examples:
the Golden Gate Bridge
the Grand Canyon
a Russian song
a Shakespearean sonnet
a Freudian slip

In any Grammar book it shall be clear that sentence #1 – "Your favourite Japanese restaurant" – is the correct usage.