The Commonwealth website avoids the problem by only referring to it by the full name:

Kingdom of eSwatini

Wikipedia, usually careful about details like this, uses Eswatini as the English transliteration and eSwatini as the Swazi version:

Swaziland (/ˈswɑːzɪlænd, -lənd/) or Eswatini (Swazi: eSwatini [ɛswa̯tˈiːni]), officially the Kingdom of Eswatini since 19 April 2018

However, the section on Government and politics starts with:

eSwatini is an absolute diarchy with constitutional provisions and Swazi law and customs.

I would say, whichever you choose, be consistent.

The Commonwealth website avoids the problem by only referring to it by the full name:

Kingdom of eSwatini

Wikipedia, usually careful about details like this, uses Eswatini as the English transliteration and eSwatini as the Swazi version:

Swaziland (/ˈswɑːzɪlænd, -lənd/) or Eswatini (Swazi: eSwatini [ɛswa̯tˈiːni]), officially the Kingdom of Eswatini since 19 April 2018

However, the section on Government and politics starts with:

eSwatini is an absolute diarchy with constitutional provisions and Swazi law and customs.

So it seems they can't really make up their minds about it.

I would say, whichever you choose, be consistent.