Inverting mean-value mapping in exponential family

Suppose I have an exponential family and fit it to data using maximum
likelihood. Finding the parameters corresponding to max.likelihood
solution is easy in mean-value parametrization while it can be quite
hard in natural/canonical parametrization (example below). My question
is -- if I only care about a probability of a single novel point, can I
do without finding the natural parameters somehow?

Example:
Suppose my domain is X_1,...,X_n \in {0,1}^n
Suff. statistic is \phi(x_1,...,x_n)=x_1 x_2 + .... + x_n-1 x_n + x_n
x_1
The complexity of finding MLE in natural parametrization grows
exponentially in n

But if I only cared about the value of this max.likelihood fitted model
at a single point y_1,...,y_n, I don't really need the whole MLE, so
could this be done faster?

.