I don't understand who owned the home or sold it and who is reporting it. Your question is very confusing.
If a person receives a home as a gift, their starting cost basis is the same as the cost basis of the giver, which is usually what the giver paid. Then the new owner can add the cost of permanent improvements to their cost basis.
You file your taxes on the honor system. I can't stop you from guessing at the cost. But if you are audited, the IRS does not have to award any adjustment that you can't prove with adequate records. The IRS can take off the adjustments you can't prove and recalculate the capital gains tax in their favor, plus interest and maybe a penalty if the change results in a substantial underpayment of tax.
I don't know why the bank would have records of the home improvements. If money was borrowed on an equity loan, bank records showing the amount borrowed won't prove the money was spent on improvements, instead of something else.
