No. Do not do this.

Me being lazy and my cat being dirty, I decided to give this a go. After the third flush I opened the lid to see if my cat was clean, only to find that she wasn't there. After panicking for a while, I looked again and she still wasn't there. I opened the sewage pipes outside to see if she was stuck, but no. Later that day I put in a call to Werribee Sewage Farm ("Fresh sewage 'cause we grow our own") and yes, they had filtered out a very irate cat of the same dimensions and colour as the one I described. I explained to them that, no, the cat wasn't an unwanted Christmas gift and, yes, I did want her back. If anything, she was even dirtier than when I put her through the cleaning process.

Remember: Flushing the toilet is to get rid of things. Don't put things you want to keep in the toilet bowl.