How to prototype with an 8-pin 'daughter board'
Get the mating 2x4 pin connector, then make up a cable that ends in 8 wires you can stick into the breadboard anywhere you want.  You could even solder them to a DIP header, which can be easily plugged onto the breadboard.

How are you going to connect to this 2x4 pin header in the final product?  If you're going to plug it into a socket as intended, then use another socket to make a breadboard adapter.  If it's a one-off and you plan to solder to the 2x4 pin header, then you might as well solder wires to it for breadboard use now.

I don't understand the requirement to not solder.  Soldering is simple, quick, reasonably reliable, and can also be easily undone.  If you don't want to get solder on the header pins, what about soldering to the bottom of the pins on the other side of the board, assuming the header is thru hole?  If it's SMD, then there should be pins running out from the connector you can solder thin wires to.  At .1" spacing you've got lots of room, so it should be easy.