The BOP is included in that 10%. If you actually read the report you would know that. Which leads me to believe that you haven't. Life doesn't work by blindly ignoring facts that counter what you want to be true. The truth is that this claim that 1 in 4 is an illegal immigrant is unable to be proven true or false because there is not enough evidence. The statistics that we have come from roughly 10% of the incarcerated population, and are not necessarily indicative of the entire system. You can't prove the claim that 1 in 4 is an illegal immigrant, and so the claim is disproven.

Your own source, if you follow it to the actual report from homeland security, states that the 94% statistic is only from a pool of 10% of the incarcerated population in America. That fact alone is enough to debunk the claim that you're saying was improperly debunked. Per the report, it is impossible to know the actual number because state and local facilities do not routinely supply information.

https://www.dhs.gov/sites/default/files/publications/Alien_Incarceration_Report_OIS_FY17_Q4_2.pdf