Konstantin,
The connection seems to work fine.
Can you give us a link to the specifications of the level converter used? If a slow level converter for I2C is used, the signal may be attenuated and not propagated.
It may work without a level converter.
Please measure the INPUT and OUTPUT voltages at the LM1084 terminals and let us know.
Is the battery fully charged?
The operating voltage of the FT1190 is 6V or less, so please supply 5V to the + pin.