I have done a FHA 203K rehab. loan but the work was much more extensive vs. a basement.

I think the question you may wish to ask yourself - what is the appraised value with the basement work completed vs. without it? And what is the LTV in both scenarios?
If your appraisal doesn't incorporate the basement as a significant value add (which i doubt it will) - I would suggest you apply for a regular mortgage covering the home plus an equity loan and use the HE to repair the basement. It will have shorter terms thus a lower interest payment over the life of the loan.

In simple terms - 15 yrs from now you don't want to pay interest on a basement which is outdated if you still own the place.

Who is she? RE agent? yours? seller? I would put is a bid for it being unfinished, because the seller or who she is, does not know how you want it done. After you move in, you may have completly different ideas. One thing, just make sure it is safe--that is how long has it been unfinished, what type of flooring, and all permits have been pulled and approved. It could be a problem if the permits have not been finalized and signed off.
Also is their plumbing, electric and those permits have been signed off.
Good luck

Get several quotes on the basement finishing before you proceed. There may be unforseen issues, like ceiling height, windows, plumbing, electric, HVAC, etc. that you have not considered. Our experience was that quotes came in all over the board. We had quotes from licensed/insured contractors for 45K - 18K for the same basement project.

our county has new egress rules that mandate a second way to get out of the basement before you can put a bedroom down there. It changed a lot of remodeling plans and budgets!