You've got to search for an alternate location, consider schools, work, commutes. Neighbors. Etc.
I can see major arguments for managing rent as a social function.
Though that depends heavily on supply and demand factors as well.
Economic incentives should encourage local, suitable (even for apartments) housing in an area to be approximately equal to the number of jobs in that area (then pad up slightly for BIG if that's also an 'employer').
1) If a single mother/father only spent 1 hour a week taking care of their kid, they would surely be punished for neglect. So we are willing to literally coerce people who choose to be parents into working at least 52*18...938 hours. At a minimum wage of $5.35, that is at least $5,000 of coercion.
2) If we have a workable basic income, then someone shouldn't find themselves in the desperate position of having a medical procedure done to make ends meet unless we don't have universal healthcare.
I know of 3 small children in my own life whose parents collect welfare, and are meth addicted.
Maybe for the first 20 years of the program every child affords a parent some additional BI based on the child's age. Raising a child when health care is inexpensive(my vision of BI does include pretty much free health care), job's are optional, and schooling is free makes the most expensive times to be early in the child's life when they are constantly outgrowing their clothing and beds and using up diapers at an impressive rate.
Eventually the amount of BI awarded becomes sufficient for most family situations(Star Trek post-scarcity style) or other factors like perfect birth control allow us to remove the child BI bonus.
Imagine the minimum income for children was diverted into services. So the government buys free clothes, school breakfasts+lunches, and medical insurance. This decreases the cost for a poor family to raise a child which allows you to give less money per child, and reduces the incentives for child farming.
This sounds like the edge case a non-parent would bring up. People who make poor life choices make them for far less insightful reasons that time-value calculus.
It would take an unbelievable amount of money for me to have another kid. It would be cheaper to buy my retirement.
If you don't give basic income for kids, then people with a lot of kids won't be able to live.
Of course, you can adjust the amount between the too extremes of full basic income and no basic income for kids, but just because you have a target standard of living and give the parents money to meet that for the kids, doesn't mean the parents aren't going to turn around and spend that money on themselves. They might not care that their kids are in poverty as long as they get the drugs/alcohol/new car/fashionable clothes or whatever.
As you already stated, bureaucratic welfare programs don't work, so what is the solution here?
I've never seen this discussed, and it seems like a major oversight of the basic income proponents.