As long as you did your market analysis correctly either deal will net you $100,000 to $500,000 appraised equity on day one.  

Is it possible to get an independent appraisal done on the property ?

Since the property is there, I can only do investor loans in AL, but from a consumer standpoint you can do Conventional, FHA, VA, and USDA as a renovation loan where you take the cost of the property plus the cost of the renovations as your purchase price up to the value of an "As-Repaired" appraisal.

Assuming you got a conventional loan and put 25% down, based on an appraised value of $365k you potentially could pull an additional $112,500 of equity.

While obviously, tax assessments and home appraisals are two different things, they aren’t supposed to be wildly divergent, right?

The appraiser also called me to ask these things.

I would pay off the HELOC you guys used for the renovations, get your property appraised again, and then use the equity to invest in the next property. 

Get an appraisal etc...

Purchase price was $180k, appraisal was $255k for a previous buyer that fell out of contract, initial ARV estimate was $320k - 340k.

Get an appraisal and 3rd party building inspection and you should do fine.