Do you plan to assemble a transcriptome? If not, you don't have to remove them. When assigning reads to feature of a GTF, they will not be counted anyways. 20% does not sound too bad. If you did not use a kit at all, it would be like 95%, so as long as you get sufficient reads mapping on target for your analysis, I would not care about them. Still as a remark, BWA is not a splice-aware aligner. You might consider using HISAT2 or STAR for RNA-seq alignments.

I have used SortMeRNA before and worked very well for me.

There is a program named BBSplit that you can use to filter and remove your undesired sequences

There are a number of tools specifically for rRNA reads removal. Recently, we also developed a rRNA reads detection method named RiboDetector (https://github.com/hzi-bifo/RiboDetector), and in the article we benchmarked different computational tools: https://academic.oup.com/nar/advance-article/doi/10.1093/nar/gkac112/6533611. RiboDetector is the most computationally efficient and most accurate software for rRNA reads removal. It can be used out-of-the-box without any database:

• GPU mode:

  ribodetector -t 20 \
  -l 100 \
  -i inputs/reads.1.fq.gz inputs/reads.2.fq.gz \
  -m 10 \
  -e rrna \
  --chunk_size 256 \
  -o outputs/reads.nonrrna.1.fq outputs/reads.nonrrna.2.fq
  

• CPU mode

  ribodetector_cpu -t 20 \
  -l 100 \
  -i inputs/reads.1.fq.gz inputs/reads.2.fq.gz \
  -e rrna \
  --chunk_size 256 \
  -o outputs/reads.nonrrna.1.fq outputs/reads.nonrrna.2.fq