BME280: SPI modes and power consumption

I have a BME280 connected to a ATtiny85 using SPI. There are 10k pullups because the module I bought can be used in SPI and I2C.

As I don't have an accurante milliamps meter, I'm powering both using a 5V source via a LED. This gives me 5V - 1,4V = 3,6V. This is borderline, but it's the best I have. By the brightness of the LED, I have a clue about the current consumption.

So I read the sensor, go to sleep and on wakeup, read again... Everything works except the power consumption during sleep is high.

When I use SPI mode 00, the LED is always bright with short flickering when reading the sensor.
When I use SPI mode 11, the led is off and only flashes when reading.

Is the BME's power consumption different using the different modes?
I can't find any clues in the datasheet.

It could very well be that I somehow partially burned the sensor while testing/trying/... (=learning).

PS I think the excess current goes via the SCK pin.