Rules-as-written, they must pick one subrace of elf.
Rules-based answer: Pick one or the other.
According to Mordenkainen's Tome of Foes, the shadar-kai are a subrace of elf. This excludes half-elf and human as possible offspring. A half-elf is defined as the offspring of an elf and a human, while a human is a different race from elf entirely.
There are no specific rules for the offspring of two different subraces of elves. However, Player's Handbook, p.23, defines the following rule for playing an elf:
Ancient divides among the elven people resulted in three main subraces: high elves, wood elves, and dark elves, who are commonly called drow. Choose one of these subraces.
Mordenainen's Tome of Foes additionally adds three subraces:
At the DM's discretion, you have access to more sub-races for elf characters, in addition to the subraces in the Player's Handbook. When you choose the subrace of your elf, you can choose one of the following options: eladrin, sea elf, or shadar-kai.
Rules-as-written, then, the player must pick either the statistics of a high elf or a shadar-kai elf. Anything else, such as some homebrew mix of the two, must be ruled upon by the DM.
Lore answer: Canonically undefined
The lore of the shadar-kai as elves is thus far restricted to Mordenkainen's Tome of Foes. This is because shadar-kai were not canonically elves prior to 5th edition. They were described in 4th edition as humanlike, rather than elves.
In other words, the description of the shadar-kai in Mordenkainen's Tome of Foes is the sole source of lore on that race as regards their offspring with a high elf. That work does not define any such offspring, meaning that it is undefined.