Thank you for your interesting question.
When I first read your post I thought you were asking about the liquid, but upon re-reading it I'm of the opinion you want to know about the container for the liquid. Just in case, I will hedge my bets and answer with both in mind.
Firstly, I should explain that I irrigate so the 'bag' in question is actually a sleeve. This can be washed out via the tap but, before leaving the loo to do so, I rinse it around using one of the containers that previously had toilet cleaner in. The reason I like this method is that it has a pinpoint spout which, when the container is squeezed, shoots out a jet of water that can be accurately guided to where I want it to go.It also holds 750ml of water which is enough to do the job. Once all the output is washed away, I then go to the sink and rinse it again.
Now for the liquid I finish off with. Well! this has ranged from deoderant sprays, mint mouthwash, various oils or anything else that is availablefor nothing or cheap to buy, which smells better than the previous contents of the bag.
When the sleeve is stained, one of the things I found that will clean it and make it smell reasonable is the stuff they steralise babies bottles with. It used to be called 'Milton' but nowadays Sainsbury's do their own cheap version. I also use this to clean and steralise the other equipment I use when it becomes stained.