First time using a Raspberry 3B+ for 1.8v SPI BIOS flashing, I have some questions first
This level converter has a "A" side and a "B" side. What I think I understand at the moment is the "A" side gets the 3.3v pins and the "B" side gets the 1.8V.


These converters have a higher voltage side and a lower voltage side.  The 3v3 side should be connected as the higher voltage one.  Which that is in your board is up to you to figure out.

Note that 1v8 may be marginally out of spec for some of these converters; it will probably work (did that once for I2C in a phone) but not certain.  You will want to operate at a slow clock speed regardless, because you are using high value pull-up resistors rather than push-pull drivers as would normally be used for an SPI bus - they are really meant for I2C at &lt;= 400 KHz.


  Does this mean I also have to connect up a 3.3v power supply to VCCA and a 1.8v power supply to the VCCB on this logic level converter board?


You need those two supplies connected to their respective sides (whichever those are), fortunately the PI gives you the 3v3 one - just shut the pi down before making the connection.


  Also I see the BIOS chip has a tolerance of "1.65to 1.95V", couldn't I use a couple of AA or AAA batteries to power this chip?


No, because one cell would be too little and two would be too much.  Granted, you might get away with one absolutely brand new alkaline, but it's not a great idea.  An adjustable regulator with a 1.25v or so adjust voltage can be used to make 1v8 supply from a 3v3 one.