Your probability changes with $t$ but as Michael said, you don't know how. linearly or not ? It looks like a model selection problem where your probablity $p$ :
$p=\Phi(g(t,\theta))$ may depend on a highly non linear $g(t,\theta)$ function. $\Phi$ is just a bounding function that guarantees between 0 and 1 probabilities.
A simple exploratory approach would be to try several probits for $\Phi$ with different non linear $g()$ and to perform a $g()$ model selection based on standard Information Criterias.
To answer your re-eddited question:
As you said using probit would imply numerical solutions only but you may use a logistic function instead :
Logistic function: $P[\theta(t+1)] = \frac{1}{1+\exp{(\theta(t)+\epsilon)}}$
Linearized by : $ \log{\frac{P}{1-P}} = \theta(t)+\epsilon $
I'm not sure how this can work under Kalman filter approach, but still believe that a non linear specification like $\theta(t+1)=a t^3 +bt^2+ct + d$ or many others without a random term will do the job. As you can see this function is "smoth" in the sense that it's continous and differentiable. Unfortunately adding $\epsilon$ would generate jumps of the resulting probability which is something you don't want so my advice would be to take out $\epsilon$.
Logit probablity: $P[Coin_{t+1}=H | t] = \frac{1}{1+\exp{(\theta(t))}}$
You already have randomnes in the bernoulli event (Markov Chain) and you are adding an additional source of it due to $\epsilon$. Thus, your problem could be solved as a Probit or Logit estimated by Maximum likelihood with $t$ as explanatory variable. I suppose you agree that that parsimony is very important. Unless your main objective is to apply a given method (HMM and Kalman Filter) and not to give the simplest valid solution to your problem.