What I would try when facing such a problem:
1°) Minimize $\lambda\int p'(t)^2\ dt +$ the sum of informations $p(t_k)\log (1/p(t_k))$ if heads at $t_k$, $[1-p(t_k)]\log (1/[1-p(t_k)])$ if tails.
2°) Chose $\lambda$ using (a suitable form of) cross-validation.
3°) Keep searching how people did before.