Does a greedy strategy exist for this instance of the Bin Packing Problem?
No, a greedy approach would not guarantee a solution for this problem (regardless of whether this solution is optimal or not). To see why, proceed by reduction. Suppose that you could actually find a solution for your problem in polynomial time using a greedy approach. Then, you could use your algorithm to solve in polynomial time the decision version of the Bin packing problem, which is known to be NP-complete (and, therefore, unlikely to be solvable in polynomial time).

How does this reduction work?

The decision version of the Bin packing problem involves deciding whether a certain number of bins (for example, 9) is optimal. If your hypothetical polynomial-time greedy algorithm finded a solution for 9 bins and failed to find a solution for 8 bins, then the answer for the decision problem would definitely be yes, and no otherwise.

Greedy is actually good.

All that being said, to obtain an approximate solution, a greedy approach is, probably, the best heuristic: sort the items in decreasing order of size and insert them one by one into the first bin that has room for it. This heuristic is called first-fit decreasing. The main appeal of this heuristic is that we pack the big items first and hope that the little ones fill up the spaces.