Your calculation is not right.

Under ordinary income tax, your long term capital gain cost basis is $1.

Under AMT liability, your first year cost basis is $1, and second year cost basis is $10.

For the second year, the difference is long term capital gain rate and AMT rate.

However, it depends on your whole picture to actually get the credit. If for the second year, you have many deductions or very high income, the higher cost basis in AMT calculation does not do any good for you, you lose the credit for ever.

In fact, it is not sth unusal in tax. You may lose short term capital loss deduction if you have wrongfully timed wash sale, etc.