I would recommend doing the alignment directly with bbsplit from BBTools with ambiguous2=toss and you are done.

While using bbsplit is a painless way of doing this once, if you do want to follow your original line of thought you can do the following.

1. Split human/mouse reads into separate BAMs using ideas from:
   How To Split A Bam File By Chromosome
   Extract ONLY chromosomes 1-22 from bam file - removing extraneous chr annotations Hopefully you have unique chromosome names from mouse and human otherwise nothing is going to work.

2. Once you have the BAMs. you can samtools collate | samtools fastq to get fastq-format reads. See the options for those commands to deal with secondary mappings etc. Step 1 and 2 can possibly be combined via a single pipe

3. Extract fastq headers by something line grep "@M01923:976:00000000" file.fq (use zgrep if your files are compressed). Remove @ at beginning of the header at the same time. example below

   zgrep @first_part_of_headers file_R1_001.fastq.gz | sed 's/^@//g' > fastq_headers_R1

4. Use filterbyname.sh from BBMap suite to find reads that are present in one or other file.