In a properly done wrap you take title. The deed transfers to you at closing, so you're the owner of record and permits get pulled in your name like any other purchase. The seller's underlying mortgage remains a lien against the property and the seller remains personally liable on that note, and your wrap note is secured by a second instrument in the seller's favor. Two liens, one owner, you.
That means the renovation is yours to do, subject to whatever the wrap note says. Many seller-carry notes include a clause restricting material alterations without consent, or requiring the lender to be named on the builder's risk policy. Read yours. If it's silent, you still want the seller's insurance requirements checked, because the underlying lender is mortgagee on the hazard policy and a 40k project usually needs a rider.
On protecting the improvement money: you can't eliminate the acceleration risk, because it lives in a contract between the seller and a lender you're not party to. You can shorten your exposure. Negotiate a right to cure the underlying directly and offset against your wrap payment, get an authorization letting you request payoff and status from the senior lender, and put a specific remedy in the wrap for what happens if the senior loan is accelerated, including whether the seller has to make you whole or subordinate. Whether a particular remedy is enforceable depends on your state's law, so have an attorney there draft it rather than adapting language from a course.
The risk you haven't named is your own resale. Buyers and their lenders will see the senior lien on title, and clearing it at your sale requires the seller's cooperation and an accurate payoff. Deals stall there. Build a payoff mechanism into the note now.
