It is a hard thing to do, and in my experiance I thought the meat tasted bad cause I was attached to them and played with them when they were young so I had ended up giving the meat away to my friends (talking about meat rabbits right now) but I've help process goats, rabbits, chickens, sheep, even pigeons.
When we went to a lady once to buy a boer to kill and put in freezer, her husband asked us how we can do that, my brother answered it helps not to think about it so my advice is to:
1. Not to think about it just do the job well.
2. Feed them, water them, clean there living area and get out don't sit there and watch them or play with them just do the jobs you have to do to keep them alive till the date when they must die.