Here are two (almost three) comments that you might find useful. I don't know the answer to your question in general unfortunately.

First, Autissier proved that for abelian varieties over $\overline{\mathbb Q}$ with good reduction, the Faltings height of $A$ equals the theta height of $A$ plus some contribution at infinity. See the main theorem on p. 1 of his paper *Hauteur de Faltings et hauteur de Néron-Tate du diviseur thêta*. So the difference of these two heights can be computed by a contribution at infinity (as you desire).

Secondly, if you consider only Jacobians then you can say a little bit more.
In fact, if $A = Jac(X)$ is the Jacobian of a curve $X$ over $\overline{\mathbb Q}$, then the arithmetic Noether formula (due to Faltings and Moret-Bailly, see for instance Section 1.6 in https://www.math.leidenuniv.nl/scripties/RdeJong.pdf) states that $$12 h_{Fal}(X) = \omega^2(X) + \Delta(X) + \delta_{Fal}(X) - 4g\log(2\pi).$$ If you assume the Jacobian to have good reduction over $\overline{\mathbb Q}$, then this doesn't imply $X$ has good reduction over $\overline{\mathbb Q}$. Nevertheless, it might be instructive to look at Jacobians of curves with good reduction over $\overline{\mathbb Q}$. So let's assume that $X$ has good reduction over $\overline{\mathbb Q}$ as well. Then, by definition of the discriminant $\Delta(X)$, the Noether formula reduces to

$$12 h_{Fal}(X) = \omega^2(X) + \delta_{Fal}(X) - 4g\log(2\pi).$$

You now see that the Faltings height of $A$ is given by $\omega^2(X) +\delta_{Fal}(X) - 4g\log(2\pi)$, as $h_{Fal}(A) = h_{Fal}(X)$. The self-intersection of the dualizing sheaf is never zero (Bogomolov conjecture), so that the Faltings height will always involve $\omega^2$ (in some sense). You can go a bit further now and rewrite the above formula in terms of the height of the Weierstrass divisor and purely analytic contributions. To do so, use Prop. 2.5.2 in loc. cit. The end product is an expression for the Faltings height of $A$ in terms of the height of the Weierstrass divisor and the analytic invariants $R(X)$ and $\delta_{Fal}(X)$. Now your question reduces to:

Can the height of the Weierstrass divisor with respect to the relative dualizing sheaf endowed with the Arakelov metric on the minimal regular semi-stable model of $X$ be computed in analytic terms solely?

Thirdly, in the case of CM abelian varieties of abelian type (a case you most likely are interested in) there are also formulas relating the Faltings height to periods due to Colmez. I can also comment on this if you'd like.

Note: I used some notation and terminology that I find convenient. Don't hesitate to let me know if elaboration needed.