Alternative to grout for thin fill on flooring?

Hi,

I have a client that's asked me to tile a hallway using large 300x300 tiles made to look like nine individuals 100x100 'tiles' (I hope that makes sense?!). The line of light grey 'grout' inbetween the nine sections of individual 'tiles' on the 300x300 tile is max 2mm so it would be ideal to continue that slim grout line inbetween the 300x300 tiles but I'm concerned that's too thin for a floor, where movement might cause the grout to crack and it'll also be tricky to get a good solid fill. Can anyone offer any advice? Is there a suitable alternative to traditional flexible grout? I've contacted the manufacturers about it but they've not responded.

Also, the job is tiling over wooden floorboards in a victorian terrace so I need to put some backing boards down first so I'd welcome recommendations for what brand/type to use.

Thanks