@Jennifer Guertin talk with a smart mortgage professional and find out if you can use the 5% down conventional if you sell both of your current properties. That would make your bids more competitive without forcing you to use the FHA, which is admittedly much trickier to use on any kind of cash flowing multi unit.
You might be able to do a 1031 exchange on the investment property. This would allow you to get into a much more interesting opportunity like a small apartment building. You could sell your primary tax free if you have lived there for 2 of the past 5 years.
@Jennifer Guertin , There's some interesting options that you've got.
1. I think you've lived in the house you're in for 5 years so a sale of that would be tax free. That's a big deal. Then a simple purchase of a MF to house hack would be a great option.
2. You're past the timelines for the former primary so that would not be tax free but you could use a 1031 exchange to sell it and buy another investment property. You could still use a 5% owner occupant loan to buy up to a 4 plex (I believe - mortgage folks correct me if I'm wrong) with your 1031 as long as the 3 units you rent equaled the value of the property you sell.
3. My inclination would be to sell both - Sell the former primary and 1031 into a MF that you house hack. Sell the current primary tax free and use those proceeds to supplement the purchase of the new investment property to bring down your ltv. This will dramatically raise your NOI and you can then use that for accelerated savings for your next property.