First, some monopolies are monopolies forever - network and utility monopolies.
Second, you still have to deal with aggregate monopolies - ie when one societal class has monopoly pricing power over another. You cannot break up these as they are already broken up.
The prime example is real estate and land market. Landlords have monopolistic pricing power over tenants. Price for rent is not governed by landlord's costs but by what the tenant can bear.
If you inject UBI to the economy then tenants can suddenly bear more and landlords can and will rise rent prices convincing others that their cost increased so it's justified.
It's not the inflation of produced goods which would kill VAT or similarly funded UBI but rent and monopoly good inflation which by definition can't be competed away.
Therefore before you can implement UBI you have to deal with monopolies first.
Hint1: Fix it by implementing monopoly based taxes such as Land Value Tax and natural resource auctions. You will raise more than you need and will also abolish labor taxes stiffling productivity.
Hint2: Georgists (economical movement after guy named Henry George in 19th century - who coined the term "Progressive") were discussing UBI ideas already pretty extensively and understood these implications. They realized the monopoly problem and made it central to their teaching. Only instead of UBI they called it a Citizen's Dividend.