I had something similar happen, I was trying to refinance with PenFed for one of their great rates. The appraisal that came in was exactly the same as when we purchased the house almost 2 years previously even though we had added $80K landscaping (waterfall in the front yard)/fencing/RV pad/basketball court on 3/4 acre and updated the master bath shower from a glass door type to a no door walk in travertine shower.
Upgrades that while I didn't expect to see a dollar for dollar increase but maybe 50%?
Couldn't do the deal and had to eat the appraisal.
We were told to try #1 and appeal the appraisal, which we did but all the appraiser says is "I think my numbers are right".
It wasn't a 3rd party going over the appraisal and appeal, it was just sent back to the original appraiser and they said yay or nay, when comparing to the different comps.
In your scenario, I would definitely pay the extra $6k and do the deal.
I don't want to spend my entire life at work. I deserve more. - Want2retire aka W2R