You're correct that a weakness of biomaRt is that you can't match the results with a specific part of the query, it just dumps back everything that matched at least once, and querying with each region separately isn't very quick if you have thousands of regions.

If you want to do this in R, then I would recommend the ensembldb package, which is a nice interface to query Ensembl data on your local machine without requiring web queries.

The example below should get what you want. It assumes your bed file is saved in /tmp/example.bed which you should change:

library(rtracklayer)
library(ensembldb)
library(EnsDb.Hsapiens.v86)
library(stringr)
library(dplyr)
library(magrittr)

## Making a short name for our geneome database
edb <- EnsDb.Hsapiens.v86

## import a bed file in a GRanges object,
## then split this into a list of indivdual ranges
regions_list <- import("/tmp/example.bed") %>%
    split(., factor(.))

## define a function that returns the names of any genes
## falling within a single GRange region
## Names are in comma separated string
getGeneNamesFromRegion <- function(x, edb) { 
    genes(edb, filter = GRangesFilter(x))$gene_name %>%
        paste(collapse = ",")
}

## apply the function above to each region in our list
genes_by_region <- lapply(regions_list, 
                          FUN = getGeneNamesFromRegion, edb = edb)

## break regions back into chrom, start, end
## and combine with the comma separated gene names
results <- str_split(names(genes_by_region), pattern = ":|-", 
                     simplify = TRUE, n = 3) %>% 
    set_colnames(c('chrom', 'start', 'end')) %>%
    as_tibble() %>%
    mutate(gene_names = unlist(genes_by_region))

And the results look like:

> results
# A tibble: 5 x 4
  chrom start    end      gene_names                                          
  <chr> <chr>    <chr>    <chr>                                               
1 1     2665698  4665777  TTC34,ABC14-1188822O7.1,RP11-740P5.2,RP11-740P5.3,A…
2 1     10391436 12391516 PGD,RP4-736L20.3,APITD1-CORT,APITD1,CORT,DFFA,RP5-1…
3 1     15106832 17106911 KAZN,TMEM51-AS1,TMEM51,C1orf195,RP1-21O18.3,RP5-903…
4 1     23436536 25436616 ASAP3,E2F2,RP1-150O5.3,ID3,MDS2,RP11-223J15.2,RPL11…
5 1     23436576 25436656 ASAP3,E2F2,RP1-150O5.3,ID3,MDS2,RP11-223J15.2,RPL11…

You could use BEDOPS bedmap to map gene annotations to regions.

First, preprocess your BED file of regions (e.g., regions.unsorted.bed) to make them UCSC-like and sorted:

$ awk '{ print "chr"$0; }' regions.unsorted.bed | sort-bed - > regions.bed

Adding chr is useful, because annotations from other groups use that chromosome naming convention. Having all the input files use the same naming scheme helps make set and map operations work correctly.

Next, grab annotations and turn them into a sorted BED file.

For example, if you are working with hg38, you could use Gencode v28 annotations and BEDOPS convert2bed:

$ wget -qO- ftp://ftp.ebi.ac.uk/pub/databases/gencode/Gencode_human/release_28/gencode.v28.annotation.gff3.gz \
    | gunzip --stdout - \
    | awk '$3 == "gene"' - \
    | convert2bed -i gff - \
    > genes.bed

Finally, map genes to regions:

$ bedmap --echo --echo-map-id-uniq --delim '\t' regions.bed genes.bed > answer.bed

The file answer.bed will contain your regions (--echo). The final column of each line will have a list of unique gene IDs of genes that overlap that line's region (--echo-map-id-uniq).

If you want the full annotation, you can use --echo-map in place of --echo-map-id-uniq. There are other --echo-map-* options; see the documentation or run bedmap --help.