Earlier quoted context omitted.
Land value tax would be the way to go. Regulating the rent would be about as useful and effective as existing rent control schemes.. With a land value tax, the rent would still go up, but the increase would mostly be recycled back into tax take. (And you can use that tax take to eg finance (part of) the UBI.)
If you intend to tax away all the advantages of providing additional housing supply, I’d expect smart landlords to not increase housing supply.
A land value tax is levied on the value of the unimproved land in a specific plot. Whether you leave the land fallow or build 30 storeys of luxury condominiums on it, the amount of dollars you have to pay in taxes doesn't change.
So the (additional) housing supply is completely untaxed.
What is taxed however is the following:
If the town gets a new highway or airport, and that raises land prices, that gets taxed. Same for a UBI.
(To pre-empt the next discussion: it's not completely trivial to separate the value of the building from the value of the land. But it can be done. Insurance companies do it routinely. And there are a variety of strategies to make the assessment that are hard to game or cheat.)