Early release in this situation is unlikely but not impossible, and a furlough is the more realistic avenue to pursue. Some facilities will authorize a supervised furlough for an inmate to be present during a family medical emergency or death, but these are granted rarely and almost always under strict conditions. At medium custody, the bar is higher than it would be for a minimum-security inmate. When they do occur, the inmate is typically accompanied by two corrections officers for the duration, the visit is time-limited, and the entire arrangement has to be approved at the administrative level.
The way to pursue this is for your brother to immediately request a meeting with his case manager or counselor and present the situation formally. Documentation matters: hospice enrollment records, a letter from her medical provider, and anything else that establishes the severity of her condition and the immediacy of the situation. The request needs to go through the proper chain within the facility, not just a verbal ask to a floor officer.
From the outside, you can contact the facility's warden's office or the case manager directly, explain the circumstances respectfully, and ask what the process is for requesting a compassionate furlough in a situation like this. Being polite and organized gives the request a better chance than an emotional appeal.
Four months remaining and a parent in hospice is a genuinely compelling circumstance. It does not guarantee anything, but it is worth formally requesting without delay, because these decisions take time and time is limited.