OK, I've computed the safe check probabilities for the chess and Gothic Chess pieces on both an 8x8 and a 10x8 board. As Ed mentioned, finding a nice formula for the safe check probability of a Bishop is hard, but since we are really only interested in the values on an 8x8 and on a 10x8 board, it suffices to compute them without bothering to find a nice formula. Of course, a Chancellor and Archbishop will never set foot on an 8x8 board, but their safe check probabilities on such a board are still of interest for the purpose of comparison. Normalizing the piece values so that a Knight on an 8x8 board has a value of 3.00, we obtain the following (SCP = "safe check probability", PV = "piece value").

(8x8 Board)
Piece.................SCP.................PV
-----................-------..............-----
Knight.............12/144..............3.00
Bishop.............13/144..............3.25
Rook...............24/144..............6.00
Archbishop......25/144..............6.25
Chancellor.......36/144..............9.00
Queen.............37/144..............9.25

(10x8 Board)
Piece................SCP.................PV
------...............------...............-----
Knight..............110/1580.........2.51
Bishop..............133/1580.........3.03
Archbishop.......243/1580.........5.54
Rook................249/1580.........5.67
Chancellor........359/1580.........8.18
Queen..............382/1580.........8.70

As Ed points out, the piece values go down (according to this theory) on a 10x8 board. The other interesting point is that this theory asserts that a Rook and an Archbishop have essentially the same value. In fact, because of the Knight's loss of range on a 10x8 board, the Archbishop appears to have slightly less value than a Rook on a 10x8 board. Of course, it would be naive to assume that this the whole story. It would seem that an Archbishop would be worth considerably more during the early part of the game when there are few open files and the Rooks haven't had a chance to enter the fray, however, it could be that a Rook and Archbishop have about the same relative value in the endgame (just as a Rook can often hold its own against two pieces in the endgame). I'd be interested to hear Ed's opinion on this. I suspect he added the mysterious "extra parameter" to the expression for the safe check probability for an Archbishop because, based on his playing experience, he didn't believe that a Rook was equal (or even slightly superior) to an Archbishop. In the end, playing experience is worth more than point values assigned on a purely combinatorial basis.