You're selling a possession problem with a house attached, and that's priced separately from the house. Start there, because it changes every other decision.
The sister with eleven years of occupancy and no rent is the whole deal. Depending on your state she may be a tenant at will, a licensee, or in a handful of states have some claim worth arguing about, and the removal process, timeline, and cost differ enormously by state and often by county. Some places that's a 45-day process. Some places it's nine months with a judge who hates the filing. You cannot underwrite this until you know which one you're in, and that's a question for a local attorney before you spend anything else on it.
On your three options. Getting her out inside your due diligence window is the one that sounds cleanest and is usually the worst, because you'd be directing a possession action on property you don't own, with no standing, and the seller is the only one who can act. If he's willing, the deal becomes: he delivers vacant, and your contract closes on that condition. Sellers in this situation frequently agree and then don't do it, so build a walk-away, not a hope.
Selling it occupied is genuinely doable, and the buyer pool is small. People who buy occupied distress underwrite the cleanout and the eviction as line items, and they'll take $15k to $25k off for the possession risk plus whatever the cleanout runs, which on floor-to-ceiling is often $8k to $15k on its own. Your $10k fee is not the constraint. Your $130k might be.
Closing it yourself is where the money is and it's a different business. You'd need the $130k plus holding costs through an unknown possession timeline, and you'd own the sister's situation personally.
Run the numbers on cash-for-keys before anything else. A few thousand dollars and moving help, negotiated by the seller while he still holds title, resolves more of these than any legal process and does it faster. If he won't have that conversation with his own sister, that tells you what the rest of the deal looks like.