What is the plural possessive form of the word "British"? I believe British is both singular and plural. Is that right? I could get around the possessive problem by using "The British empire's...", but how would I do it with just the one word?
It appears as if you're asking about this meaning of the word British
the natives or inhabitants of Britain
In that case, it's a mass noun, meaning you cannot say "a British," in the same way that you cannot say "a water" or "a sugar." In other words, it's a singular noun, but it represents multiple things.
Janus Bahs Jacquet has pointed out that if British is a noun, it must be plural, since we would say, "The British are coming" not "The British is coming." Furthermore, "what [British] really is is an adjective modifying a deleted head noun, and as such, it can also be singular, and it can also be countable, though a generic head noun (one) will usually be supplied in that case."
The possessive form would be the British's, but this looks and sounds a little awkward, so it's probably best to avoid it. You could use the British people's instead.
This ngram, the British's, the British people's suggests that the British's popularity is rising and the Bristish people's is falling. This is surprising, but it might suggest that there's more support for the British's than I realized.
I should note that if you want to talk about British people, the following would also be a correct plural possessive:
The Britons' food preference.
From Merriam-Webster's definition of Briton:
1 : a member of one of the peoples inhabiting Britain prior to the Anglo-Saxon invasions
2 : a native or subject of Great Britain especially : ENGLISHMAN
Briton is often shortened to Brit.
Using this noun can help to distinguish it from British, which could be confused with either the country or the adjective. This noun is also countable and, unlike British, you can say a Briton or a Brit.
So, it may be preferable in some contexts.