Another thing, if you're getting 10% down, that doesn't factor in closing costs and any other costs associated with financing (processing fee, underwriting fee, origination fee, appraisal fee, etc).

@Taylor L. so there is no exceptions even if the Appraiser comes with the Fair Market rents?

It might be worthwhile asking an appraiser what work might get it marked as resolved in a bank appraisal so that it can market to the mortgage-borrowing segment of buyers at closer to full value.

Double check them and figure on them going up especially if the purchase price is significantly more than the current appraised tax value.  

So Better has preapproved us and said we don't need an appraisal on the new house we are purchasing.

Work with a good realtor and mortgage lender, and utilize the experience to learn about what it takes to find a home, qualify for a mortgage to buy the home, qualify the home for the mortgage through the inspection and appraisal processes and the closing process, will help position you to successfully do things more independently in the future. 

Be prepared that if you are bound and determined to buy in the current market you could very well end up with the house appraising for less than you are offering, and would have to come up with the difference CASH (if this is a surprise, PM me for a full explanation). 

When you close, there's no bank, no points, no appraisal - you just close with your own cash. 

My lender doesn’t seem sure about my question if my signed lease agreement supersedes the rental appraisal.

The way to get a house right now is to offer all-cash and waive appraisal.