I need some advice in how to go about fixing my ongoing "dissapearing grout" issue in the kitchen floor.

To make the story short, when the leg column was initially being installed, but porcelain floor (3/8") tile cracked (fault of the installer, not of the tile) and it had to be replaced. Due to the fact that the new one only had 48 hours to dry (it was glued in with thinset), it seems that this time was not enough because when the column was put in, I guess the screws that hold it in place "tilted" the new tile by a fraction. This has in effect makes the tile move (sway) maybe by a couple of milimeters when someone steps on it. We have tried to re-grout the spot 3 times now, but due to this slightest of all movements, the grout eventually cracks and within 5-6 weeks almost dissapears.

It has been suggested to me that to fix this (without redoing the whole job from scratch, as it is impossible since the countertop breakfast table sits on that leg) that I should try one of the two following solutions:

1) I should mix the grout and right before applying it, I should add a little bit of CEMEX glue and apply that combo directly into the gaps. This way, it will keep the grout color and the cemex will prevent the grout from carcking again.

2) I should just buy a beige-colored CAULK (no idea if acrylic or silicone ?!?), and just fill-in the gaps.

What should I do to make this a more "permanent" solution? I want to whatever I do to be able to withstand cleaning of this "very much used" floor with a WET Swiffer Sweeper cloth, and not having these gaps re-open up.

All suggestions are very much appreciated!