Air carrier Captain reported an unstable approach to BNA where the First Officer was below the glide slope; resulting in a GPWS warning. The First Officer corrected the vertical path and continued the approach to a landing.

Narrative

At time of release the TAF at destination +/- 1 hr was indicating VFR condition even the METAR prior to 2.5hr flight was indicating 4SM OVC (Overcast) 007. This release did not require a landing alternate. During the flight; ATIS reports were continually reporting weather in the vicinity of 1 1/2SM and OVC 002 to 0003. During the flight; I (CA) reached out to dispatch via ACARS to ask about possible alternate destinations in case the weather further deteriorated at BNA. No response was ever received from dispatch. The CA with concern for fuel on board briefed the FO (PF) that if there were any issues getting into BNA we would need to immediately divert to HSV (approximately 90 miles away) with minimum and possibly emergency fuel. At the start of the approach;the weather had improved to 2SM and OVC 006. However; the CA remained concerned with fuel and possible alternate destinations in the case of a missed approach. The approach remained normal and stable until approximately 300' AGL. At which point the CA called out a transient increase in descent rate which was corrected. The CA then called the FO was low on the PAPI at which time the Glide Slope EGPWS warning sounded. The FO leveled the descent and recaptured their descent; and completed a landing in the touchdown without any further issue.Upon landing; I as the CA realized that the situation warranted a go around call due to an unstable approach. Continued concerns about being able to break through the weather at a suitable landing field with the fuel available adversely affected the decision to call for the go around. After landing; the flight concluded with no further incident.