There are a host of Web sites dedicated to "deep searching" on the Internet. This is also called accessing the invisible Internet.
I clicked on CompletePlanet - The Deep Web Directory, which claims over 70,000 searchable databases and specialty search engines: "A comprehensive listing of dynamic searchable databases. Find databases with highly relevant documents that cannot be crawled or indexed by surface web search engines."
I searched for "Jewish genealogy, family history." In the initial list of hits appeared Family History in India. The site is designed "to help people trace their British, European and Anglo-Indian family history in colonial India. Much of the website is devoted to British and Christian families, but this series of pages refers to non-British family history."
Just for the fun of it, I investigated to see if there were any indications of members of the tribe in the site's general resources.
Among the offerings was a general Index to Bengal marriages (with thousands of individuals). Though this is not - at first glance - a Jewish source, it contained some surprises. The indices list volume and folio of the records and where the records are kept.
Using common Jewish surnames, I did a quick search. There is no guarantee that these individuals were indeed our brethren, but perhaps your lost branch is among them.
Index to Bengal Marriages:
1860: Cohen Frederick and Schoenerstedt Emilie
1864: Cohen Lionel A and Hawkins Eliza
1887: Goldstein Walter H. and Curton Agness A.
1890: Goldsteen Mary and Duncan James
1893: Cohen Richal and Hutchison Fredk.
1893: Goldstain Ellena and Hamilton Jas.
1893: Moskowitz Chaja and Vlastelicich Vincenzo
Index to Calcutta Marriages:
1795: Goinstine, Jacob and "Mary"
1795: Judah, Abraham and Blair, Anne
1782: Sinai, Joseph and Danston, Maria
Index to High Court Records, Calcutta 1818: James Levy
Indian Mutiny Medal – British Forces:
Private Lewis Levy - 14th Light Dragoons
Sergeant Major Samuel Levi - 1st Battalion 20th Foot (East Devonshire)
British Army Casualty List 2nd Afghan War 1878-1880:
Henry Cohen - Drummer - 66th Foot - Killed in Action - Maiwand 27-07-1880
There is a search engine which finds "sounds like" as well as exact matches. For example, entering STEIN brings up STEEN and STONE.