How do Bayesians interpret $P(X=x|\theta=c)$, and does this pose a challenge when interpreting the posterior? I have seen the post Bayesian vs frequentist interpretations of probability and others like it but this does not address the question I am posing.  These other posts provide interpretations related to prior and posterior probabilities, $\pi(\theta)$ and $\pi(\theta|\boldsymbol{x})$, not $P(X=x|\theta=c)$.  I am not interested in the likelihood as a function of the parameter and the observed data, I am interested in the interpretation of the probability distribution of unrealized data points.
For example, let $X_1,...,X_n\sim Bernoulli(\theta)$ be the result of $n$ coin tosses and $\theta\sim Beta(a,b)$ so that $\pi(\theta|\boldsymbol{x})$ is the pdf of a $Beta(a+\sum x,b + n - \sum x)$.
How do Bayesians interpret $\theta=c$?  $\theta$ of course is treated as an unrealized or unobservable realization of a random variable, but that still does not define or interpret the probability of heads. $\pi(\theta)$ is typically considered as the prior belief of the experimenter regarding $\theta$, but what is $\theta=c$?  That is, how do we interpret a single value in the support of $\pi(\theta)$?  Is it a long-run probability?  Is it a belief?    How does this influence our interpretation of the prior and posterior?
For instance, if $\theta=c$ and equivalently $P(X=1|\theta=c)=c$ is my belief that the coin will land heads, then $\pi(\theta)$ is my belief about my belief, and in some sense so too is the prior predictive distribution $P(X=1)=\int\theta\pi(\theta)d\theta=\frac{a}{a+b}$.  To say "if $\theta=c$ is known" is to say that I know my own beliefs.  To say "if $\theta$ is unknown" is to say I only have a belief about my beliefs.  How do we justify interpreting beliefs about beliefs as applicable to the coin under investigation?
If $\theta=c$ and equivalently $P(X=1|\theta=c)=c$ is the unknown fixed true long-run probability for the coin under investigation: How do we justify blending two interpretations of probability in Bayes theorem as if they are equivalent?  How does Bayes theorem not imply there is only one type of probability?  How are we able to apply posterior probability statements to the unknown fixed true $\theta=c$ under investigation?
The answer must address these specific questions.  While references are much appreciated, the answers to these questions must be provided.  I have provided four Options or proposals in my own solution below as an answer, with the challenges of interpreting $P(X=x|\theta=c)$ as a belief or as a long-run frequency.  Please identify which Option in my answer most closely maps to your answer, and provide suggestions for improving my answer.
I am not writing $P(X=x|\theta=c)$ to be contemptuous.  I am writing it to be explicit since $P(X=x|Y=y)$ is not the same thing as $P(X=x|Y)$.  One might instead be inclined to write in terms of a sample from the prior and use an index of realizations of $\theta$.  However, I do not want to present this in terms of a finite sample from the prior.
More generally, how do Bayesians interpret $P(X=x|\theta=c)$ or $P(X\le x|\theta=c)$ for any probability model and does this interpretation pose any challenges when interpreting $P(\theta=s|\boldsymbol{x})$ or $P(\theta\le s|\boldsymbol{x})$?
I've seen a few other posts tackle questions about Bayesian posterior probability, but the solutions aren't very satisfying and usually only consider a superficial interpretation, e.g. coherent representations of information.
Related threads:
Examples of Bayesian and frequentist approaches giving different results
Bayesian vs frequentist interpretations of probability
UPDATE:
I received several answers.  It appears that a belief interpretation for $P(X=x|\theta=c)$ is the most appropriate under the Bayesian paradigm, with $\theta$ as the limiting proportion of heads (which is not a probability) and $\pi(\theta)$ representing belief about $\theta$.  I have amended Option 1 in my answer to accurately reflect two different belief interpretations for $P(X=x|\theta=c)$.  I have also suggested how Bayes theorem can produce reasonable point and interval estimates for $\theta$ despite these shortcoming regarding interpretation.

I think part of the problem is that there are some notational problems in the question, and a degree of people talking past each other due to having different backgrounds/positions, so I'll go through the question trying to understand what was meant.  I will be happy to be corrected if I am wrong and will edit the answer until we understand each other.
The first issue is what does the author mean by $P(X=x|\theta=c)$?  I think this is intended to mean the probability that the random variable $X$ has the value $x$ if the parameter of the model, $\theta$ has its "true" value, $c$.

How do Bayesians interpret θ=c, the probability of heads? θ of course
is an unrealized or unobservable realization of a random variable,

This is a problematic line for me as $\theta$ is not a random variable, but a parameter of the model.  If we knew what $c$ was, we would just set $\theta = c$ and there would be no need for a prior or a posterior.  But we don't know the optimal value of the parameter, so what do we do?
The traditional Bayesian approach is to contsruct a prior for the unknown parameter value, $\pi(\theta)$ that represents what we know about the parameter a-priori (which may be very little).  If we want to know what values of $\theta$ are plausible, given our prior and our data point, $X = x$, then we use Bayes rule, giving
$p(\theta|X = x) = \frac{P(X = x|\theta)p(\theta)}{P(X=x)}$
Notice I have written $P(X=x|\theta)$ rather than $P(X = x|\theta = c)$.  This is because we are not interested in a single number telling us the probability of a head.  We want to continue representing our knowledge in the form of a distribution of relative plausibilities of all possible values of $\theta$.  Representing knowledge in the form distributions, rather than point values is fairly central to Bayesianism.
IF we wanted to give a single number representing the probability of a head, then we might take the mode of $P(\theta|X=x)$ or the expectation of $\theta$ with respect to $P(\theta|X=x)$.  But asking how Bayesians interpret $\theta = c$ seems meaningless, it is just setting a parameter of our model to a particular value.

For instance, if P(X=1|θ=c)=c is my belief that the coin will land
heads, then π(θ) is my belief about my belief, and in some sense so
too is the prior predictive distribution P(X=1)=∫θπ(θ)dθ=aa+b. To say
"if θ=c is known" is to say that I know my own beliefs. To say "if θ
is unknown" is to say I only have a belief about my beliefs.

This seems very confused.  In the case of flipping a coin (a Bernoulli trial), then $P(X=1|\theta=c) = c$ is a tautology as the parameter of a Bernoulli distribution is the probability that $X=1$, so this equation only holds when the parameter of the distribution is equal to its true value. But we don't know the value of $c$, so Bayesians wouldn't encounter this.  $\theta$ is a parameter of a model, $c$ is it's true value, what more could there be?
$P(X=1|θ=c)=c$ is not my belief that the coin will land heads, it is the true probability that it will land heads.  It can't be my belief as it relies on me knowing the correct value of the parameter $\theta$, but I don't.  This means that "then π(θ) is my belief about my belief," is incorrect, because the premise was incorrect.  It is just your belief about the relative plausibilities of different values of the parameter $\theta$.

To say "if θ=c is known" is to say that I know my own beliefs.

No, this would be equivalent to saying that you know the true value of the parameter $\theta$, so it is just saying the prior should be a delta function centered on $c$.  It is just a direct statement of your prior belief/state of knowledge.

To say "if θ is unknown" is to say I only have a belief about my beliefs.

Again, this is incorrect because the premise at the start of the paragraph was false.  It just means you don't know the true value of parameter $\theta$ so perhaps a flat prior distribution on the interval 0 to 1 would be appropriate (encoded as a Beta distribution for convenience).
I think I'll leave it at that for now, adding more is likely to just be further talking past eachother, so I will wait for @GeoffreyJohnson 's comments/corrections.
