Husk, 14 13 10 bytes

-1 byte thanks to Dominic van Essen
-3 bytes thanks to Jonathan Allan's trick

►Ofo=▲¹Σπ3

Try it online!

Explanation

►Ofo=▲¹Σπ3
        π3   cartesian product of 3 * input
  fo         filter by composed function
     ▲¹      max of input
     =  Σ     equals sum
►            max by
 O           sorted

Husk, 14 13 bytes

-1 byte thanks to Dominic van Essen

◄§-▼▲fo=▲¹Σπ3

Try it online!

Explanation

◄§-▼▲fo=▲¹Σπ3
           π3     cartesian product of 3 * input
     fo          filter by composed function
       =▲¹Σ      max of input equals sum
◄§               min by composed binary function
  -▼▲            max - min

Husk, 14 13 bytes

-1 byte thanks to Dominic van Essen

◄§-▼▲fo=▲¹Σπ3

Try it online!

Explanation

◄§-▼▲fo=▲¹Σπ3
           π3     cartesian product of 3 * input
     fo          filter by composed function
       =▲¹Σ      max of input equals sum
◄§               min by composed binary function
  -▼▲            max - min

Husk, 14 13 10 bytes

-1 byte thanks to Dominic van Essen
-3 bytes thanks to Jonathan Allan's trick

►Ofo=▲¹Σπ3

Try it online!

Explanation

►Ofo=▲¹Σπ3
        π3   cartesian product of 3 * input
  fo         filter by composed function
     ▲¹      max of input 
    =  Σ     equals sum
►            max by
 O           sorted

Husk, 14 bytes

◄§-▼▲fo=▲¹ΣΠR3

Try it online!

Explanation

◄§-▼▲fo=▲¹ΣΠR3
            R3   repeat input 3 times
           Π     cartesian product
     fo          filter by composed function
       =▲¹Σ      max of input equals sum
◄§               min by composed binary function
  -▼▲            max - min

Husk, 14 13 bytes

-1 byte thanks to Dominic van Essen

◄§-▼▲fo=▲¹Σπ3

Try it online!

Explanation

◄§-▼▲fo=▲¹Σπ3
           π3     cartesian product of 3 * input
     fo          filter by composed function
       =▲¹Σ      max of input equals sum
◄§               min by composed binary function
  -▼▲            max - min