Level shifter default high problem
The level shifter in your picture is very likely based on this schematic:



The pull-up resistors will keep the lines HIGH while there are at high impedance. That's what you're measuring.

However, as soon one of the ports is connected to a non-hiZ OUTPUT pin, that won't be case anymore. The OUTPUT pin will then be able to drive LOW the port it is connected to, and then the other port will go LOW as well.

Conclusion: it should be fine for SPI.

EDIT:

The Amazon product page you've linked has a picture that contains a big hint too: