@Peace Lily To clarify, is the 3-plex you're wanting to purchase 100 mile or more away from your current duplex? I am in no way an FHA or duplex expert, but the reason I ask this is there's a rule (I only recently found out about) called the 100 mile rule with FHA where if you aren't moving at least 100 miles away from your last residence (and you keep it as an investment) then FHA won't consider the income from your first property. I.E. you would have to have the income, excluding rental income, to cover both mortgages.
I know that doesn't exactly address your question regarding the 3-plex being considered an investment property by FHA, have you talked with the loan officers about providing proof that your duplex is already fully rented out, to show your intentions of occupying the 3-plex? Also I agree with @Joe Jor that if you haven't already you might talk to local credit unions and see if they can help you out. Finally, if that fails you may see what the banks offer as far as low down-payment conventional home loans, I know when I was looking earlier in the year (June-July) there was an option for me to do conventional at 5% down payment.