The way I finally solved this issue was by modifying the stargazer function. Particularly, in .stargazer.wrap, which code you can find here, I added the following code:
else if (class(object.name)[1] == "betareg") { #betareg
          .summary.object <<- summary(object.name, type = "person")
inside .add.model, i.e. just after these lines of code:
.add.model <-
      function(object.name, user.coef=NULL, user.se=NULL, user.t=NULL, user.p=NULL, auto.t=TRUE, auto.p=TRUE, user.ci.lb=NULL, user.ci.rb=NULL) {
        
        if (class(object.name)[1] == "Glm") {
          .summary.object <<- summary.glm(object.name)
        }
Note that you need to create a new stargazer function with the new .stargazer.wrap. I just called it stargazer2:
stargazer2 <-
  function(..., type = "latex", title="", style="default", summary=NULL, out=NULL, out.header=FALSE,
           column.labels=NULL, column.separate = NULL, covariate.labels=NULL, dep.var.caption=NULL, 
           dep.var.labels=NULL, dep.var.labels.include=TRUE, align=FALSE, coef=NULL, se=NULL, t=NULL, 
           p=NULL,  t.auto=TRUE, p.auto=TRUE, ci=FALSE, ci.custom=NULL, ci.level=0.95, ci.separator=NULL, 
           add.lines=NULL, apply.coef=NULL, apply.se=NULL, apply.t=NULL, apply.p=NULL, apply.ci=NULL, 
           colnames = NULL, column.sep.width = "5pt", 
           decimal.mark=NULL, df=TRUE, digit.separate=NULL, digit.separator=NULL, digits=NULL, digits.extra=NULL, 
           flip=FALSE,
           float=TRUE, float.env="table", font.size=NULL, header=TRUE, initial.zero=NULL, intercept.bottom=TRUE, 
           intercept.top=FALSE, keep=NULL, keep.stat=NULL, label="", model.names=NULL, model.numbers=NULL, 
           multicolumn=TRUE, no.space=NULL, notes=NULL, notes.align=NULL, notes.append=TRUE, notes.label=NULL, 
           object.names=FALSE,
           omit=NULL, omit.labels=NULL, omit.stat=NULL, omit.summary.stat=NULL, omit.table.layout=NULL,
           omit.yes.no=c("Yes","No"), order=NULL, ord.intercepts=FALSE, 
           perl=FALSE, report=NULL, rownames = NULL,
           rq.se = "nid", selection.equation=FALSE, single.row=FALSE, star.char=NULL, 
           star.cutoffs=NULL, suppress.errors=FALSE, table.layout=NULL, table.placement = "!htbp", zero.component=FALSE, 
           summary.logical=TRUE, summary.stat=NULL, nobs=TRUE, mean.sd=TRUE, min.max=TRUE, median=FALSE, 
           iqr=FALSE) {
    
    save.warn.option <- getOption("warn") 
    options(warn=-1)
    return(.stargazer.wrap(..., type=type, title=title, style=style, summary=summary, out=out, out.header=out.header,
                           column.labels=column.labels, column.separate = column.separate,
                           covariate.labels=covariate.labels, dep.var.caption = dep.var.caption,
                           dep.var.labels=dep.var.labels, dep.var.labels.include=dep.var.labels.include,
                           align=align, coef=coef, se=se, t=t, p=p, t.auto=t.auto, p.auto=p.auto, 
                           ci=ci, ci.custom=ci.custom, ci.level=ci.level, ci.separator = ci.separator,
                           add.lines=add.lines, apply.coef=apply.coef, apply.se=apply.se, apply.t=apply.t, 
                           apply.p=apply.p, apply.ci=apply.ci, colnames=colnames,
                           column.sep.width=column.sep.width, decimal.mark=decimal.mark, df=df,
                           digit.separate=digit.separate, digit.separator=digit.separator, 
                           digits=digits, digits.extra=digits.extra, 
                           flip=flip,
                           float=float, float.env=float.env,
                           font.size=font.size, header=header, 
                           initial.zero=initial.zero, 
                           intercept.bottom=intercept.bottom, intercept.top=intercept.top, 
                           keep = keep, keep.stat = keep.stat,
                           label = label,
                           model.names=model.names, model.numbers=model.numbers,
                           multicolumn = multicolumn,
                           no.space=no.space, notes=notes, notes.align=notes.align, 
                           notes.append=notes.append, notes.label=notes.label, object.names=object.names,
                           omit=omit, omit.labels=omit.labels, omit.stat=omit.stat, omit.summary.stat=omit.summary.stat,
                           omit.table.layout=omit.table.layout,omit.yes.no=omit.yes.no, 
                           order=order, ord.intercepts=ord.intercepts, perl=perl, 
                           report=report, rownames=rownames, rq.se=rq.se, selection.equation=selection.equation,
                           single.row=single.row, star.char=star.char, 
                           star.cutoffs=star.cutoffs, suppress.errors=suppress.errors,
                           table.layout=table.layout,
                           table.placement = table.placement, zero.component=zero.component,
                           summary.logical = summary.logical,
                           summary.stat = summary.stat,
                           nobs=nobs, mean.sd=mean.sd, 
                           min.max=min.max, median=median, iqr=iqr, warn=save.warn.option))
  }