...destroying (it's not murder if they're not sentient)...
More interestingly, the EMH isn't exactly an individual, especially if copied thousandfold. Instead of destroying him, our heroes would have a range of options that wouldn't apply to a live doctor. They could for example merge the inconvenient 10,000 doctors into one; the "averaging process" might amount to losing perhaps 0.003 doctors' worth of identity, as most of the material would be common already between the clones.
(logically) it's a violation of the Prime Directive to steal/acquire technology through timetravel.
Our heroes didn't object to using Borg technology, even though it was for all practical and theoretical purposes "stolen", too. And one might argue it was also acquired through "time travel", because the Borg represent future technologies from the UFP point of view...
In backing up the EMH, I think our heroes would have used any means necessary and justified it as being a life-saving measure for a fellow hero creature.
Timo Saloniemi