What if you merge the .bed files, and quantify counts within the peaks?
Do you get a high correlation in the quantifications?
In your case, I think there is some bigger difference (for HTH3), but that is what I would probably check if the peak counts were similar but in different positions (like for the other antibodies).
For ATAC-Seq data, I found it helpful to use the --local option to increase the alignment rate. You can also run Picard and get an idea if the insert distribution looks different with the different alignments.
However, for ATAC-Seq data, the alignment rate was very different for default BWA-MEM and Bowtie2. Is the alignment rate also different for your samples, or do you have a similar alignment rate and a different number of peaks?
Also, for Histone modifications, I used the HOMER findPeaks with -style histone . However, I don't think it changed things as much as you described (for HTH3).
Finally, is your total reads similar in your replicates? That can also have an effect on the number of peaks called (and ATpoint also mentions read depth and read count comparisons).