Selling a house involves lots of taxes
If the house is owned by an entity you control, then think twice before selling it.
A transfer from one entity to another - even if both are the same hand - involves a bunch of overhead costs - when we sold a $75,000 parcel this was almost $10,000, and I was very mad at my partner, because the property was owned by an LLC that owned only that, specifically so we could just sell the LLEC and avoid all that overhead. I don't know what the costs would be on a $1M property (it's not proportional).
However, trusts also may pay taxes, so you should look carefully at the tax situation both ways, and see what is most favorable to you. This is work for a tax accountant, not yo-yo's on the internet. Given the value of the trust - a million dollars - that is sensible (even if you don't feel like a millionaire).
You cannot reset an assessment with fake sales.
The house is 100% paid off, so there are no monthly payments. I don't own the house, it's owned by a trust, so as trustee I can sell it and then buy it myself at a much lower rate than what it was appraised at at time of death.
Wow, maybe you need some professional help being trustee if you're dreaming up two-bit schemes like that. Obvious scams are obvious. Everyone will "see right through it" - the IRS, the courts, and the local property tax assessor. That will not work, but it will establish you as an unreliable schemer, which will get everything else you've done examined with a magnifying glass.
Your scheme seems to rely on using the assessment that once was valid, to wedge your sister out of the 25% stake. By your logic if the house fell from $400,000 to $300,000, all that cash loss came out of your sister's 1/4 and your 3/4 is still full-value at $300,000. LOL yeah that doesn't work. When the value of the house falls, it's still 75%/25%. Your 75% shrinks too so in that example your sister's stake would be $75,000 and yours $225,000.
The house value is what your sister says it is
... and that's how it is. You can't force a seller to do something they don't want to do.
So whatever the sister thinks the house is worth, 1/4 of that is the price you must pay, and that's that. There's no workaround, short of something illegal like extortion. BUT THAT WOULD BE INSANE. There is a sad misconception in American life that money is the most important - no. What Suze Orman says:
People first. Then money. Then things.
Family is forever and family goes first. That's that. Yes, of course you have had your rivalries. Look at France and Germany - they've had a quarrel or two - but when stuff matters, they stand together in NATO and the EU. You do not want to estrange your sister - lose that "much bigger than friends" family bond -- over thirty pieces of silver. Seriously. I can't state this strongly enough.
You seem to be on the road to perdition here, so let me remind you what "perdition" looks like. Your sister hates your guts. Your sister lawyers up and tears apart all your activity as trustee, looking for fraud or misappropriation, so they can expel you as trustee and get someone else as trustee. They can tie up that house and keep you from a) cashing out or b) living there for as long as it takes until you break. Then they buy you out for a fraction because you just want the nightmare to be over and never see the inside of a law office ever again. That's what "estranged" means. And it'll be your fault, because you went down that road.
So quit taking this family rivalry so seriously. You're on the same team. You need to be reasonable, and I really recommend you be kind.
But the trust can get a mortgage (with you co-signing).
And if you're only borrowing 25% of property value, the qualifications for such a mortgage won't be too high. Once the trust has the cash in hand, the trust can pay your sister the 25% trust value, in cash. Now you control a trust with a mortgage, and YOU need to make sure that mortgage gets paid or you'll lose the house.
On the other hand... the sister isn't in a rush
But consider this as well. You're the one clamoring to sell. Not your sister. Sounds to me like your sister isn't in need of the quarter million, and is happy to leave it sitting as equity in the house.
So here's a crazy idea. Rent the house from the trust. You have the trust say that since you're the trustee and 3/4 heir, it would be silly to make you pay 100% rent only to disperse 75% of it back to you. Therefore you only need to pay 25% rent which goes to your sister of course.
Now you're renting for peanuts and your sister is happy.