What is the best way to form a confidence interval for a weighted sum of expected responses in a GLM? Suppose you have a generalised linear model (GLM) with link function $g$ so that the true expected responses in your model (with $n$ data points) are:
$$\mu_i \equiv \mathbb{E}(Y_i | \mathbf{x}_i) = g^{-1}(\mathbf{x}_i \boldsymbol{\beta})
\quad \quad \quad \text{for } i = 1,...,n.$$
Now, suppose you have a set of known weights $\omega_1,...,\omega_n$ and you want to make an inference about the weighted sum of the expected responses:
$$\mu_\omega \equiv \sum_{i=1}^n \omega_i \mu_i =  \sum_{i=1}^n \omega_i \cdot g^{-1}(\mathbf{x}_i \boldsymbol{\beta}).$$
This kind of problem arises in certain applications where you want to form some kind of "standardised" or "weighted" expected response using a set of exogenous weights.  In most models of interest, the link function $g$ is nonlinear, so its inverse $g^{-1}$ is also nonlinear.  Thus, the inference here is (usually) about a nonlinear function of the parameter vector $\boldsymbol{\beta}$.  Presumably this means that forming a confidence interval involves use of the asymptotic normal distribution for the coefficient vector, combined with the delta method to deal with the nonlinear transform.  I am aware of some general ways this could be done (e.g., delta method, simulation, etc.), but I'm not sure what is "best practice" here.

Question: What is the best way to form a confidence interval for $\mu_\omega$?  How is this implemented using a glm model in R?  (An example for a logistic regression would be lovely.)
 A: What you're asking is essentially the general form of how do I get a confidence interval for the mean of a function broadcast over a sample. This should help you out. You're already 95% of the way there.

Thus, the inference here is (usually) about a nonlinear function of the parameter vector $\beta$. Presumably this means that forming a confidence interval involves use of the asymptotic normal distribution for the coefficient vector, combined with the delta method to deal with the nonlinear transform. I am aware of some general ways this could be done, but I'm not sure what is "best practice" here.

This is precisely what you should do if you want a confidence interval. If the data are independent, then it's fairly easy to derive. After you compute your Delta method distribution, you're just multiplying by a vector. So, $$ \sqrt{n}\left(w^T g^{-1}(X\beta) - w^T\mu_\delta \right) \sim N\left(0, w^T \Sigma_\delta w \right)$$
You have to be careful about how you define the covariance matrix you use in your Delta method approximation. Do you want a confidence interval for the mean (in which case you want to use the $\Sigma = \Sigma_\beta$ from the coefficient estimates)? Or do you want a prediction interval for the quantity (in which case you'll want to use $\Sigma = \Sigma_\beta + \sigma^2 I$ combining the prediction uncertainty as well)?
You should be able to derive the necessary quantities for logistic regression with this. You seem to know the Delta method already. You know $g^{-1}$ for logistic regression, $\frac{\exp(X\beta)}{1 + \exp{X\beta}}$. You can get the necessary covariance matrix from glm. It's just a matter of plugging it in.
