Bluetooth transfer problem with big files
I suspect that this is a String class heap-fragementation issue (again). To test this you could sending a fixed string buffer (vector of characters, "Hello world" :). Do not use String and SD/FAT in the test. Just send a large number of lines with dummy data. 

If this works replace the String usage with a character buffer. Fill it from the SD file (buffer size at a time) and write to Serial. 

Cheers!