This is a twist on a classic DP problem, and it admits DP solutions.  After we
convince ourselves that various greedy approaches do not work (can you find
the counterexamples for "take largest" and "take smallest"?), we can try to
find a recurrence that we can use.  The key is that unlike in the classic
problem, where the order of objects does not matter, in this case we want to
process them in sorted order, because that allows us to write a more nuanced
recurrence.  If you look through the reference solutions, you will find
multiple variations on the recurrence.  Here is one of them:

Let dp[i][j] = -1 if it is not possible to achieve a total size of exactly j
with some subset of the first i blocks, or c if the largest value for which it
is possible to achieve a total size of exactly j with some subset of the first
i blocks, including all blocks of size at most that value, is c.  We can then
iterate over the final row of this DP table and check to see whether the
smallest of the remaining blocks in each scenario can fit.  Like with the
classic DP problem, this can be compressed into a single row with some care;
doing so results in the solution in BadPacking_AN.java.

You can peruse the other solutions for a variety of other formulations.