FAIRE for non-standard cultivar, mapping to reference, MAPQ

Hi,

I have FAIRE-seq data from two cultivars: reference B73 and less known S68.
I assumed I can align reads from both cultivars to B73 reference sequence, which is standard.
I used bbmap with default settings.

I've seen, that MAPQ>10 read filtering is frequently applied.
The problem is, that (I think) too low percentage of reads pass this filter. It is ca. 50-60% for B73, which I assume should be mapped very well, and ca. 40-48% for S68.

I have folowing questions:

1. What settings, for example for bbmap or bwa mem are good for alignment of chromatin-level data from non-standard cultivar to reference genome?
2. What MAPQ threshold should I apply?
3. I haven't filtered fastq files for base qualities (looks good in fastqc after adapter removal). Should I do this filtering?

Here are MAPQ plots from bamqc
b73 cultivar

s68 cultivar