Hi
Most of the Tates in the Longtown area were buried at Kirkandrews on Esk rather than Arthuret. There is one gravestone there, which is mostly illegible (apart from the names Robert, John and Thomas) but on the reverse is:
Interred here Sarah TINNISWOOD who died at Longtown August 6th 180- aged 97 years.
From the parish register
1806 Jul 29 Tinniswood Sarah aged 94 widow of Arthuret.
Since there are other Tate gravestones nearby, its possible that this is for your Tate family, as there are no other Tinniswoods buried at Kirkandrews on Esk, and that Sarah Tinniswood is a relative of Elizabeth Tinniswood.
Robert Tate married Elizabeth Tinniswood at Arthuret in 1781:
Jan 30th 1781 Marr. TATE Robert Tate of Longtown in this parish, Batchelor & Elizabeth Tinninswood X of the same place, Spinster by Banns (from the Arthuret BTs)
it might be worth checking out Kirkandrews on Esk for the burials of some of Tate families there, if your Robert and Elizabeth were buried there then their ancestors may have come from that parish as well.
cheers
robbiesmum