Ok, let's spell it out with a basic example
You said we are going to give $12 billion to the poor
There is $747 billion of credit card debt in the United States. Much of that comes from lending in the past to the poor to finance their basic needs.
$747 billion > $12 billion
It is possible that all of 1 year's basic income entirely goes toward paying off credit card debt. In this scenario:
1) The poor receive a much needed benefit from basic income: they get to pay off their credit card bills.
2) There is no change in demand on basic goods whatsoever.
3) This significant paying down of debt leads to consumer interest rates going down, which further helps the poor.
So here is one scenario that is a counterexample to your claim that this can "any other change hurts the poor". This is an example which helps the poor.
You are forgetting that people have the freedom to do what they want with their money. When prices go up for one thing, they spend their money in other ways.
Healthy ways for prices to change:
1) The price change is stable, rather than runaway. It happens once, and that's it, but we continue to pay basic income year after year. This is expected, if we keep the money supply fixed.
2) The price change eventually leads to an increase in wages for the poor, which have been stagnant for decades.