It should grow with inflation, but not necessarily automatically with CPI.

My quite tentative, back of envelope analysis:

If it grows at a fixed rate, then when inflation outpaces it we will see the real value shrink. Since a larger BI is likely to be inflationary, this shrinkage would lead to some deflation (or at least reduced inflation), which will grow it again. Hopefully this would act as some stabilization and give the Fed a hand at targeting a particular rate of inflation, and a predictable rate of inflation seems to be what's best.

If there was a UBI, then the yearly raises everyone voted themselves would be a significant cause of inflation.

No, because the basic income would be paid for by a tax. Tax revenues increases with inflation too.

And that increase in tax revenue will be offset by an increase in prices, which in turn will be offset by an increase in the UBI, ...

To expound on the point, absent other economic factors, an inflation-adjusted UBI pretty much guarantees that next year's inflation will be greater than this year's inflation.

I think you are falling for Zeno's paradox. There exists an equilibrium with a constant percentage rate of tax redistributed as a lump sum that will not affect inflation at all. If the lump sum is larger than the tax revenue, there will be inflation, and if it is smaller there will be deflation. Just look at the total money supply. Prices in general are not directly affected because this is a redistribution of income.