Besides filling up some dish-washing detergent in a travel-sized container on my own...I want to know if any companies sell pre-packaged packets or tiny bottles of organic dish-washing liquid that kills bacteria etcetera?
I think your best bet is what you suggest in your first sentence: get a travel-size container of your own and fill it up with washing up liquid. For completeness sake travel-sized containers must be of a volume ≤ 100ml / 3.4 fl.oz.
The first thing that comes to mind is to recycle existing containers. Off the top of my head I am thinking of reusing one of those hand-sanitizer containers of the type that people commonly carry with them when winter falls in fear of being contaminated with the flu. Being pocket-/purse-sized they might fall under the travel-sized category too. The other option is to purchase one of the many travel-sized container kits one can find (see here for several Amazon listings), and fill one of the containers with washing-up liquid.