Why are you opposed to the Doctor having to face his own mortality, and actually face that he will die? The character wouldn't know he was indeed going to Regenerate. Lots of stories could be spun from both his belief he really could die and and also from his renewed appreciation for life after unexpectedly Regenerating
I'm opposed to it because we know that he won't die even if the character doesn't. All this sniffing and whining and eventual acceptance (unless they have him dragged away kicking and screaming) of his mortality is meaningless if the audience doesn't believe it for a second. It's the same reason why I disliked the story arc of series six, the more they tried to convince me that the Doctor would die at that friggin lake in Utah the more it annoyed me because I knew he wouldn't die, there was no doubt about it. The question was never if he would survive jut how he would weasel his way out of being shot.
Another thing is, wasn't it Ten who already did the I don't want to die part? He pretty much stated that regenerating equals death for the current Doctor, his memories survive but he's no longer the same person. I'm sure all stories about the thirteenth Doctor facing permanent death could be done with any Doctor facing regeneration with a little bit of tweaking. Just elaborate on what Ten said, make it clear to the audience that regenerating doesn't mean putting on a new meat suit, that it's scary and a definite end for the current regeneration.
Edit: Maybe they could give a Doctor some kind of gallifreyan cancer, it could be healed by conventional therapies if he's lucky, otherwise he regenerates, the companion could think that that's actually a pretty cool way to beat cancer. Have the Doctor snap back that it's not cool and that regenerating doesn't mean he'll have survived the cancer, it just means another guy remembers him. An illness would put him in a position where would be faced with a looming regeneration/death over a longer period of time, that could be used as a storyarc similar to the "no regenerations left" scenario, have him deal with his illness for an entire season and in the end he loses the battle and regenerates. That way the writer's could have the Doctor face his "mortality" without bullshitting the audience and a "Whoa, I'm the 14th Doctor!!!" non-surprise.