Another issue popped up in "How would you change the show", so let's go at it.
Any episode before "Living Witness", a senior staff meeting.
JANEWAY: "Lieutenant Torres. I understand you and Ensign Kim have been working on a unique project."
TORRES: "Yes, Captain. As you know, having only the one copy of the Doctor has long been a concern for us. We think we may have found a way to create a backup for him."
PARIS: "I thought the Doctor's program couldn't be backed up."
TORRES: "Well, we knew that it had to be possible to replicate his program, because Dr. Zimmerman made multiple copies for the fleet. He obviously didn't program them all individually."
KIM: "We found some details of the duplication process in Voyager's archived files. They were sketchy, but they've given us a starting point."
EMH: "I hope you're not planning on having multiple copies of me running around sickbay."
TORRES: "Nothing like that, Doctor. We're not sure the holo-emitters in Sickbay could even handle running two EMH's at once. This would just be an archived copy, to be used only if you were lost somehow."
EMH: "Oh. I guess that would be all right."
JANEWAY: "Nice work, both of you. Keep me apprised of your progress."
Or...
JANEWAY: "Lieutenant Torres. I understand you and Ensign Kim have been working on a unique project."
TORRES: "Yes, Captain. We finally remembered that course from the Academy called "basic computing" and how easy it is to copy a programme to a spare hard drive - turns out they did this all the time back in 2001! So yeah, turns out it is really easy to back up the Drs programme...actually we found it in the ships servers. Tom, you and Harry must feel really stupid for spending all that time trying to make one last year, eh?."