Remove all personality software, intelligence, etc and have it just as a mindless tool. It doesn't need to be anything else, after all they aren't sentient.
I don't think that's a very productive row to hoe, there, since there's no way to say that the EMH's (or any AI) aren't sentient without either A) conceding that WE aren't sentient (in which case, what the hell does sentient even mean, anyway), or B) Appealing to some anthropocentric pseudomystical appeal to authority that, frankly, has no place in this forum.
That being said, if they are consigned to using the EMH-1's for menial tasks, you'd think they'd have come up with a better solution than asteroid duty. Don't they already have robots (like the Exocomps) for that sort of work?
Though I could see the application in dangerous environments, where even 'expensive' equipment might be seen as too much or a risk to deploy, since holograms come with no material risk, as long as the mainframe the programs are stored on is somewhere relatively safe and free from damage, the holoprograms can do anything and go anywhere in the range of their emitters with no risk to themselves or their equipment, so I suppose it does make a 'sort' of sense, economically.
And yes, removing any personality protocols would probably be a good idea, if for no other reason than to truncate any potentially budding personalities or sentience emergence from arising in the first place...
Personally, I don't see that as amoral. Removing the personality from an already sentient being would be monstrous, but simply not including it to begin with would be no worse than a genetically engineered life form that didn't have higher brain functions included, but otherwise looked human. A bit ghastly, but not 'wrong' in any ethical sense.