An alternative is to refi your current mortgage balance and then get a HELOC.

I am neither a realtor nor a mortgage lender so I have no alternative conflict of interest and I don't have a training program that I can sell you. 

Does it makes sense to pass on a property you plan on holding over the foreseeable future due to not getting the FHA loan or does it make sense to close with an alternative form of financing leaving other exit strategies on the table?

Cash out refi's may increase your monthly payment - once you to do this you are locked in.Another alternative to consider is a HELOC.

Alternatively, I was considering selling my unit and using the profits to invest in two properties: a new primary residence and an investment property. 

Alternately Charge them full months rent and deposit upon occupancy and charge them prorated rent  for August but paid the next month on Sept 1.   

Alternately, you could encapsulate it so any asbestos fibers cannot escape. 

Alternatively, you can ask the seller to give notice prior to closing to the MtM tenant or, depending on the timing, the one with the lease.

If you want to mimic an appraiser's approach (wise move since their are the ones determining value) you can search up to 1 mile away and up to 6 months before to be easily useful in comparing to your subject property.If you can't find a comparable sale to your subject property then the next best alternative is to take similar properties and adjust them to match the subject property (for example, if your comparables are all 3 bed 2 bath homes and your subject property is a 4 bed 2 bath, you might ADD something like $4,500-8,000 to the sale price of the comparable properties to indicate the likely value of your property.

The alternative is - you wholesale your current deal to another investor and make enough for a 15% down + fees & costs.