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.

What I would try when facing such a problem:

1°) Minimize $\lambda\int p'(t)^2\ dt +$ the sum of informations $\log (1/p(t_k))$ if heads at $t_k$, $\log (1/[1-p(t_k)])$ if tails.

2°) Chose $\lambda$ using (a suitable form of) cross-validation.

3°) Keep searching how people did before.