My wife is concerned that we would lose most of the money in fees.One lender suggested refinancing our home to a conventional loan and using the VA loan for a multi family home to get started. 

When you factor in the cost to buy, the cost to repair, the cost of HML interest, HML points fees, inspectors, draw fees, surveys, builders risk insurance, homeowner non-occupied insurance, brokers fees, and closing cost, this stellar deal becomes a loser costing us 25K out of pocket after refinancing!

For you, you'll want to make sure that your strategy is solid, that you know you're buying at the right price, that you are budgeting rehab correctly, and that your exit strategy is very realistic (renting and refinancing). 

What bank(s) do you recommend that are friendly to the BRRRR strategy in term of refinancing?

We actually refinanced our house about a year ago and after the appraisal we were shocked that we have about $200K in equity!

Would make banks who I approach for refinancing look at the ARV and that the rents well or would they focus on my financials?

The people who are typically buying cash and refinancing after are adding value to the property through rehab.This way when they refinance based on the new ARV they can pull out all or most of their original cash to repeat.This is known as the BRRRR strategy here on the forums and there are countless posts and information about it.

Ability to extract equity through cash-out refinancing is one lever to pull but that's not a high reliability strategy due to market valuations and interest rate. 

We fixed it up and forced some equity in the property and then refinanced with a conventional mortgage to avoid mortgage interest. 

Is it possible to practice the BRRRR in some fashion after acquiring a multifamily turnkey property by fiancing 80% turning around and refinancing tlo pull out the money and buy another property?