For both Tenjiku Shogi and Taikyoku Shogi, all historical sources (historical, Western, and modern) state that the Pawn promotes to Gold General/Tokin, with no disputes whatsoever.
Since Tenjiku Shogi is basically just an enlarged version of Chu Shogi, where the Pawn also undisputedly promotes to Gold General/Tokin, there is no reason whatsoever not to include the Pawn promotion in its ludeme. As for Taikyoku Shogi, sources were obviously much harder to come by, but the ones I found all have the Pawn promoting to a Gold General/Tokin.
Tenjiku Shogi sources:
https://en.wikipedia.org/wiki/Tenjiku_shogi
https://www.chessvariants.com/invention/tenjiku-shogi
https://web.archive.org/web/200505211229...enjiku.pdf
http://history.chess.free.fr/tenjikushogi.htm
Taikyoku Shogi sources
https://en.wikipedia.org/wiki/Taikyoku_shogi
https://www.chessvariants.com/shogivaria...glish.html
Tai Shogi is a bit of a mixed bag, since its promotions are disputed, but in sources that state that the Pawn does promote, it undisputedly promotes to Gold General/Tokin.
Since Tenjiku Shogi is basically just an enlarged version of Chu Shogi, where the Pawn also undisputedly promotes to Gold General/Tokin, there is no reason whatsoever not to include the Pawn promotion in its ludeme. As for Taikyoku Shogi, sources were obviously much harder to come by, but the ones I found all have the Pawn promoting to a Gold General/Tokin.
Tenjiku Shogi sources:
https://en.wikipedia.org/wiki/Tenjiku_shogi
https://www.chessvariants.com/invention/tenjiku-shogi
https://web.archive.org/web/200505211229...enjiku.pdf
http://history.chess.free.fr/tenjikushogi.htm
Taikyoku Shogi sources
https://en.wikipedia.org/wiki/Taikyoku_shogi
https://www.chessvariants.com/shogivaria...glish.html
Tai Shogi is a bit of a mixed bag, since its promotions are disputed, but in sources that state that the Pawn does promote, it undisputedly promotes to Gold General/Tokin.