TL;DR: there are recorded games for Shatranj from the 10th century onward, but I haven't found the variant asked for by the OP.
The earliest recorded game of modern chess was 1475:
1475: Castellvi-Vinoles, Valencia 1475. The first documented chess game played with the modern queen and bishop moves; the moves were described in the poem Scachs d'amor.
Going to the predecessor of modern chess we have Shatranj:
The earliest listing of works on chess [Shatranj] is in the Fihrist, a general bibliography produced in 377 AH (988 AD) by Ibn al-Nadim. It includes an entire section on the topic of chess.
chess2u.com provides, what they claim to be, the oldest recorded chess game (10th century):
- f3 f6 2. f4 f5 3. Nf3 Nf6 4. g3 g6 5. Rg1 Rg8 6. h3 h6 7. e3 e6 8. g4 fxg4 9. hxg4 g5 10. fxg5 hxg5 11. d3 d6 12. e4 e5 13. Be3 Be6 14. Nxg5 Ke7 15. c3 Nxg4 16. Ke2 c6 17. d4 d5 18. b3 b6 19. Nd2 Nd7 20. Qc2 Qc7 21. Qd3 Qd6 22. Ndf3 Ndf6 23. Bh3 Bh6 24. Bf5 Bf4 25. Rac1 a6 26. c4 Rac8 27. c5 bxc5 28. Bxc5 Ke8 29. dxe5 Nxe5 30. Nxe6 Rxg1 31. Rxg1 Nxf3 32. Kxf3 1-0 (Black resigns)
However I haven't found any sources for the variant you mention.