There was only one movie with the Borg.
That's the Enterprise E's sickbay. So that ship is probably using nextgen tech one step beyond Bioneural Gel. Although Bev was suggesting that she had denied herself the opportunity to use an EMH for quite some time, so maybe there was an EMH on the Enterprise D?
Zimmerman asked Bashir why he didn't have an EMH. He claimed that Cardassian tech is incompatible, but maybe he was just afraid of his status being outed?
The rise of the EMH II, Andy Dick of all people, suggests that there must have been problems beyond that abrupt personality, since you gotta wonder why any one would run EMHII software on EMHI hardware?
New walls.
Ah... First Contact. I really should have remembered.
It wouldn't suprise me if Enterprise D also had an EMH, but what goes against that would be the lack of it ever being mentioned in TNG, or even holo-emitters being present in any area other than the holodecks for that matter. However, that doesn't mean necessarily that such wasn't there.
There could genuinely have been a compatibility problem with the EMH and Cardassian Technology, there was a hell of a lot of work O'Brien had to do when the Federation first got onto DS9 to get the station up to Federation spec. But you could be right, maybe the choice was made not because it couldn't be done, but because as you say, the hologram could have outed him. But that doesn't explain why he did not use an EMH after he was outed, surely after he no longer had his secret to protect, he could have used one.
As far as I am aware, all holograms need the same basic hardware, some computer storage space for their programs, and a holo-emitter, this in mind, why would EMH2 not be able to run using the same hardware as EMH1? I think it was probably the personality issue, but they could have just reprogrammed the EMH1s behavioural subroutines to counteract this problem. I guess though with any technology that's been around for a while, ways are found to improve and upgrage. I see EMH2 as the necessary next step in improvement.