Xi Chen has a theorem that says that rational curves on K3's in a linear system of dimension $>3$ are nodal. I suppose you don't need this curve to be rational, but his techniques might help you in your quest. At least his theorem tells you that you cannot expect too many cusps.
I couldn't find the paper online, I include the MathSciNet review below.
EDIT/Addendum I just realized that it might be easy to get cuspidal curves if you don't care about the genus of the curve. A degeneration of a family of nodal curves is usually a cuspidal curve. More precise statements in this regard can be found in Kollár's book on rational curves, e.g., Miyaoka's lemma. Also, Kebekus's JAG paper on singular rational curves has techniques of this kind. These are all aimed at studying rational curves on varieties, but I think some of the deformation theoretic techniques could be useful. These will (might) give you one cusp, but then you can try to consider families with $m$ cusps and try to degenrate them to $m+1$ cusps. Of course, there are obstructions to doing thiss, but you might be able to pull it off.
end of addendum
Rational curves on K3 surfaces.
J. Algebraic Geom. 8 (1999), no. 2, 245–278.
The paper is essentially divided into two parts. The first part proves existence of rational curves in the linear system |OS(d)| on a general K3 surface S in Pn (n≥3 and d>0).
The rest of the paper is devoted to the following conjecture: For n>3, all rational curves in the linear system |OS(1)| on a general K3 surface are nodal.
The conjecture is proven in the cases n≤9 and n=11 and hence justifies Yau-Zaslow's beautiful counting formula, ∑∞g=1n(g)qg=q/Δ(q) for g≤9 and g=11.
A basic ingredient in the proof is the degeneration of the K3 surface to a trigonal K3 surface, that is, a K3 surface with Picard lattice congruent to (2n−2330). The moduli space of trigonal K3 surfaces consists of countably many irreducible components (of dimension 18), and the author considers three of these in order to prove the conjecture in the cases described above. Various reformulations and generalizations of the conjecture are also introduced.