The initial, flawed response to this issue would be to construct some "massive political and logistical complexity" in order to track everyone's income and spending and family situation and lots of other stuff too and calculate their VAT refunds based on some formula. The much simpler solution is to just give everyone the same amount of money. Poor people will spend some portion of it on sales taxes, and they'll use the rest for other purposes of their choosing. Slightly richer people will spend all of the UBI on sales taxes but will come out even. Slightly richer people will be able to offset some portion of their sales taxes. Really rich people won't even notice they got a UBI.
Perhaps you really meant that "massive political and logistical complexity" is required to get anything through the legislative process? After all, the more complicated it gets, the more places lobbyists can hide loot for their employers.
TL;DR: my fear with UBI is that unscrupulous people will do unfair things like significantly raise rent to eat into other people's free money thus leaving the poor in the same spot, more or less.
I don't claim to know anything about your town, but I've lived lots of places and in general, in order to really fuck people over the bastards first have to limit their freedom somehow. Are there only certain parts of town in which the poor are allowed to live? Has public transportation been limited in arbitrary ways? Do zoning regs or HOAs enforce impractical restrictions on housing? Would it be OK if your neighbor parked a mobile home and camper behind her house for her recently-divorced sister-in-law plus five kids? What if her nephew and his girlfriend also moved out back?