How to use a Beta distribution for Bayesian meta-analysis with Brms?

I am conducting a meta-analysis using the proportion of accuracy on a specific test (0-1: mean of test scores / max test score) as effect sizes by using the Brms package in R. I would like to know how to obtain the predictions that are bounded between 0 and 1.

I started with the code something like this:

fit <- brm(proportion|se(SE) ~ task + predictor2 + (1|study/id),
      data = df,
      family  = gaussian)

SE is SE for proportion, calculated based on the SD for test scores divided by the maximum test score. task is a categorical variable referring to a type of task. predictor2 is a numerical variable that potentially relates to accuracy.

However, I soon realized that when I obtained the median of the predictions for different task types with prediction(), the predicted values sometimes go outside of 0 and 1 for some combinations of task and predictor2. Similarly, the prediction interval includes values outside 0-1.

newdata <-  
  data.frame(type = c("A",
             SE = rep(SE_median, 5),
             predictor2 = rep(0,5))

predict(res.immediate.meaning.recall, newdata = newdata,
        re_formula = NULL,  
        probs = c(.2,.8),
        sample_new_levels = "uncertainty",
        allow_new_levels = TRUE,
        summary = T,
        robust = T) %>% 

I tried to specify the Beta distribution for a response variable. However, I got an error saying

Error: Argument ‘se’ is not supported for family 'beta(logit)’

Is there any way that I can specify the beta distribution (or other alternative distribution) for this response variable?
If not, how could I deal with this issue and obtain more accurate predictions and prediction intervals?

Thank you very much for your help.

> sessionInfo()
R version 4.2.2 (2022-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Ventura 13.2.1

Matrix products: default
LAPACK: /Library/Frameworks/R.framework/Versions/4.2-arm64/Resources/lib/libRlapack.dylib

[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] bayestestR_0.13.0  tidybayes_3.0.2    brms_2.18.0        Rcpp_1.0.10       
 [5] ggpubr_0.6.0       kableExtra_1.3.4   sjPlot_2.8.12      extrafont_0.19    
 [9] MuMIn_1.47.1       pipeR_0.6.1.3      clubSandwich_0.5.8 forcats_1.0.0     
[13] stringr_1.5.0      dplyr_1.1.0        purrr_1.0.1        readr_2.1.3       
[17] tidyr_1.3.0        tibble_3.1.8       tidyverse_1.3.2    ggrepel_0.9.2     
[21] ggplot2_3.4.0      metafor_3.8-1      metadat_1.2-0      Matrix_1.5-3      
[25] readxl_1.4.1      

loaded via a namespace (and not attached):
  [1] backports_1.4.1      systemfonts_1.0.4    plyr_1.8.8           igraph_1.3.5        
  [5] svUnit_1.0.6         splines_4.2.2        crosstalk_1.2.0      TH.data_1.1-1       
  [9] rstantools_2.2.0     inline_0.3.19        digest_0.6.31        htmltools_0.5.4     
 [13] fansi_1.0.4          magrittr_2.0.3       checkmate_2.1.0      googlesheets4_1.0.1 
 [17] tzdb_0.3.0           modelr_0.1.10        RcppParallel_5.1.6   matrixStats_0.63.0  
 [21] xts_0.12.2           sandwich_3.0-2       extrafontdb_1.0      svglite_2.1.1       
 [25] timechange_0.2.0     prettyunits_1.1.1    colorspace_2.1-0     rvest_1.0.3         
 [29] ggdist_3.2.1         haven_2.5.1          xfun_0.37            callr_3.7.3         
 [33] crayon_1.5.2         jsonlite_1.8.4       lme4_1.1-31          survival_3.5-0      
 [37] zoo_1.8-11           glue_1.6.2           gtable_0.3.1         gargle_1.3.0        
 [41] emmeans_1.8.4-1      webshot_0.5.4        sjstats_0.18.2       sjmisc_2.8.9        
 [45] distributional_0.3.1 car_3.1-1            pkgbuild_1.4.0       Rttf2pt1_1.3.12     
 [49] rstan_2.21.8         abind_1.4-5          scales_1.2.1         mvtnorm_1.1-3       
 [53] DBI_1.1.3            rstatix_0.7.2        ggeffects_1.1.5      miniUI_0.1.1.1      
 [57] viridisLite_0.4.1    xtable_1.8-4         performance_0.10.2   HDInterval_0.2.4    
 [61] DT_0.27              stats4_4.2.2         StanHeaders_2.21.0-7 htmlwidgets_1.6.1   
 [65] datawizard_0.6.5     httr_1.4.4           threejs_0.3.3        arrayhelpers_1.1-0  
 [69] posterior_1.3.1      ellipsis_0.3.2       pkgconfig_2.0.3      loo_2.5.1           
 [73] farver_2.1.1         dbplyr_2.3.0         utf8_1.2.3           reshape2_1.4.4      
 [77] later_1.3.0          tidyselect_1.2.0     labeling_0.4.2       rlang_1.0.6         
 [81] munsell_0.5.0        cellranger_1.1.0     tools_4.2.2          cli_3.6.0           
 [85] generics_0.1.3       sjlabelled_1.2.0     broom_1.0.3          mathjaxr_1.6-0      
 [89] evaluate_0.20        fastmap_1.1.0        processx_3.8.0       knitr_1.42          
 [93] fs_1.6.0             nlme_3.1-162         mime_0.12            xml2_1.3.3          
 [97] shinythemes_1.2.0    compiler_4.2.2       bayesplot_1.10.0     rstudioapi_0.14     
[101] ggsignif_0.6.4       reprex_2.0.2         stringi_1.7.12       ps_1.7.2            
[105] Brobdingnag_1.2-9    lattice_0.20-45      markdown_1.5         nloptr_2.0.3        
[109] shinyjs_2.1.0        tensorA_0.36.2       vctrs_0.5.2          pillar_1.8.1        
[113] lifecycle_1.0.3      bridgesampling_1.1-2 estimability_1.4.1   insight_0.19.0      
[117] httpuv_1.6.8         R6_2.5.1             promises_1.2.0.1     gridExtra_2.3       
[121] codetools_0.2-19     gtools_3.9.4         colourpicker_1.2.0   boot_1.3-28.1       
[125] MASS_7.3-58.2        assertthat_0.2.1     withr_2.5.0          shinystan_2.6.0     
[129] multcomp_1.4-20      parallel_4.2.2       hms_1.1.2            grid_4.2.2          
[133] coda_0.19-4          minqa_1.2.5          rmarkdown_2.20       carData_3.0-5       
[137] googledrive_2.0.0    base64enc_0.1-3      shiny_1.7.4          lubridate_1.9.1     
[141] dygraphs_1.1.1.6