I have 1000 sats. I can't send the whole amount, because there's routing fees to pay to get the payment to the destination. The fees estimate was 30 sats.

So, I send 970 sats. But it still fails. Fee estimate this time was 37 sats.

So, I send 963 sats, and this time it goes through, but I end up with 9 sats left over because the fee ended up being only 28 sats.

Fee estimates aren't accurate. The fees to make a payment are calculated each time, not pre set.

The only way to get all the Bitcoin out of a channel, is to close it. It's not your channel so that's not an option for you in this case.

You can't spend all of the Bitcoin out of a channel. There'll always be some unspendable amount. It's called dust.

You're inevitably going to be leaving some dust in the mutiny wallet. Mutiny doesn't get to keep that dust. Miners will get it when the channel closes.