The expression "East Indies" was used in two senses. More narrowly, it applied specifically to the Dutch possessions that are now the nation of Indonesia. More broadly, it meant the whole of further Asia other than India, China and Japan, i.e. Ceylon, Malaya, the DEI and southeastern Asia in general. During the period in question, the Dutch ruled Ceylon and also had positions in Malaya, as well as the DEI.

The Netherlands was neutral during most of the American Revolutionary War, and the Dutch made a very good living selling munitions to the American rebels. The main Dutch depot for this trade was the West Indian island of St. Eustatius, a/k/a/ Statia. In December 1780 the British declared war on the Netherlands primarily to put a stop to this trade. In February 1781 Admiral Rodney's fleet took and sacked Statia, capturing numerous American ships and large quantites of Dutch supplies (some of which had been bought from British traders elsewhere in the West Indies).

There was naval and land fighting between British and French in India towards the end of the American Revolutionary War, in which the Dutch joined after 1781. The French, under Admiral Suffren, and their Indian ally Hyder Ali, Rajah of Mysore, had rather the better of it. (See link to Suffern wargame site). During the fighting, the British first took and then lost the port of Trincomalee in Ceylon. This is as good a place as any to put Colonel Brandon in the East Indies during the relevant time.

The UK didn't grab Ceylon from the Dutch until 1795, during the French Revolutionary Wars.