Sometimes a sarcy reply to someone like that will often shut them up...it's a bit of a reality check for them as they didn't expect to get a reply from an actual human being...other times, it's someone just frustrated in which case it definitely does come off as unprofessional...it's a fine balancing act.
Similarly, you can ignore some and they won't post again as they just wanted to vent...there are others that you'll ignore and that'll ignite a crusade in them to troll you online trying to get a rise...I've had that a lot where the individual has jumped on my reply to other people...which doesn't look great for us either. We can block them, sure! but we don't want to start blocking everyone who is mean to us.
What about the customer who said that your refurbished trains look a little basic; meaning they expected a more intense refurbishment; one of your team replied with "we think you look a little basic", that's surely un-called for, the person wasn't being abusive, just giving feedback.
With regards to being told to go and kill yourself etc, then surely that is what the block option is there for, similar to if a passenger was abusive to a member of staff at the station, they can rightly be refused travel.
Working in customer service, you can't let one person's behaviour affect how you treat the rest of your customers. You have to not be able to treat every Tweet as a personal attack. It's a case of block and report, then move on and provide excellent service to the next Tweeter. I appreciate that your job can be mentally draining, but surely if you learnt not to treat insults from customers (who do not know you, cannot see you and would probably wish no harm on you personally if they did know you) personally and let it ride over you, you might find your job easier.
Similarly, on this forum, we have a small minority of members who might treat others without the respect they would give if not behind the anonnymous comfort of fake names and avatars, we as members just have to report what we can and not take it to heart.