Ok got it finally!
Thing was to turn on SPI mode in the code, and rem out the I2C mode:
#define BMP_CS A2
#define BMP_SCK A3
#define BMP_MISO A4
#define BMP_MOSI A5
//Adafruit_BMP280 bmp; // I2C
Adafruit_BMP280 bmp(BMP_CS); // hardware SPI
//Adafruit_BMP280 bmp(BMP_CS, BMP_MOSI, BMP_MISO, BMP_SCK);