Yes, use a travel shampoo bottle for dishwashing soap. You can even do Anti-bac if you prefer. I found that there isn't much counter space for drying things in the room, but bring one of those mesh bags used for washing lingerie and you can put stuff in there and hang it to dry without fear of the recently washed cup getting knocked onto the floor.
Here is what I have learned the hard way from TSA....if you have an icepack in the bag with your yogurt, etc, it needs to be frozen solid when they inspect it. I took one with us on a trip and it was OK when we flew out. But, then, our hotel did not have a way for me to re-freeze it so I took it back through just cold, not frozen, and they took it away because it had become a "gel" by that point. So, you either find a way to refreeze it, plan to put it in your checked bag on the way back, or don't take one at all.