I haven't come across much in the way of a systematic digitization of the BEIC records and frankly the main holding at the British Library is massive (there are smaller holdings scattered throughout the US and Britain).

I would recommend surveying your secondary literature for the location of the archival material you are interested in and then using that at the basis for a search to see if anything your after has been digitized by the holding archive. Another idea might be to grab one of the plethora of finding aids that have been made for the British India Office records at the British Library (the section where many BEIC records are located).

Another possibility might be to see if any local libraries (probably university ones) have microfilmed copies of material your interested in.

Unfortunately, I think you'll find most archival information has simply not been digitized for a variety of reasons. This archive has been used extensively though, so at least much of the material has been sorted and classified and listed in finding aids-- this is definitely not always the case.