I don't think it's clear from the events that he intentionally kills his future self,
or even that he actually does - at least not with the phaser. The solution to their problem effectively erases the second Picard from existence, so he was doomed from the start.
The future Picard is a mess. He is barely recovered from being 'out of sync' with the current time frame; previously he was delirious beyond ability to even communicate at all. Sometimes, injured or unstable life-forms/starships which would normally be disabled by a low-intensity phaser blast end up dying without any intent to kill on behalf of the aggressor. The phaser blast may have simply been more than his scrambled neural pathways could handle.
In support of the previous point; he pulls the phaser directly from storage and fires without changing the settings. Typically phasers have to be set to kill; they don't just come on with 'kill' as the default setting. Indeed a phaser with the raw energy to disrupt rock faces would be a horribly dangerous weapon if it didn't require being manually set to such a setting before being fired. It's a bit like a television that doesn't turn itself down after being powered off -- annoying as hell, only substitute "incredibly deadly" for "annoying". If indeed it was intended to remember its previous setting, then surely some Ensign somewhere is about to get a negative mark on their quarterly review for stowing a weapon on a kill setting!
I'm fairly certain that no one has a chance to monitor his vitals or affirm his status before he phases out during the resolution of the episode. I'm wrong on this, he does appear quite dead after being monitored by Dr. Pulaski! However -- even if Picard had merely intended to stun him, he'd have ceased to exist anyway, which is arguably on a par with killing, depending on your view of temporal mechanics as they apply to such causality violations. (how can he travel back in time if he successfully escapes the anomaly?)
Really, there was never going to be a chance to keep the second Picard within the context of this episode. He was instantiated after the first failed attempt to escape the anomaly, which then caused the first attempt to be altered by giving them vital information about how not to proceed, so the alt-first attempt having succeeded, time-travelin' Picard can never have been created in the first place. It's a bit wibbly.