Below is the LTspice simulation of a circuit that uses the common TL431 programmable shunt reference as an accurate comparator to turn on the relay when the voltage is above the set point.
The TL431 conducts and turns on the P-MOSFET when the Ref voltage provided by the voltage divider R2-R3 from the battery voltage (Vb) goes above 2.5V.
An increase or decrease in R2 will increase or decrease the trip voltage (here shown as Vb ≈ 13.1V for the given values).
R2 can be a 50kΩ pot if you want the trip voltage to be readily adjustable.
R6 provides some hysteresis so the relay doesn't chatter around the set point, causing the relay to drop out at about 0.27V below the upper set point or Vb ≈ 12.9V here.
The LED D1 provides an indication that the relay is energized.
The transistor can be just about any standard (not logic-level type) P-MOSFET with a ≥50V and and Rds(on) of ≤0.5Ω rating.
The circuit can be built on a small perf board.
View attachment 131204