This route is not included in J. Gordon Edward's A CLIMBER'S GUIDE TO GLACIER NATIONAL PARK.
A detailed route description of route on the peak itself is included in CLIMB GLACIER NATIONAL PARK Volume 1. This guide suggests approaching over the top of Pollock, but the climb can also be approached directly from the Highline Trail - see below.
Another published reference is in the 1988 Journal of the Glacier Mountaineering Society, Route Notes, no author listed.
This description rates the climb by the Glacier Mountaineering Society ..as..III(4) SS
Bishops Cap can be approached either via the Highline Trail from Logan Pass or from the summit of Pollock Mountain via its north ridge and a traverse along the top of the Garden Wall.
From the Highline Trail, one can most easily ascend directly up a major drainage to a notch in the Garden Wall directly at the north end of Mount Pollock's north ridge. This way is no more difficult than class 2, but requires climbing some very loose scree and talus. It is a much better descent route, allowing some enjoyable scree running. One can also angle upwards directly toward Bishops Cap on class 2 and 3 ledges with much firmer footing.
Once on top of the Garden Wall, follow the goat / climbers trail to the base of the rocks on the south side of Bishops Cap. The trail ends against a short cliff.
Climb upwards to the left (west) on easy (class 3) ledges. On the west side, the easy ledges end abruptly at a steep wall. Do not continue to traverse - this would require climbing on two and three inch holds above deadly exposure. Rather, climb the approximately 20 foot high class 4 cliff straight up (climber's right). Above this cliff, look again to the right for a very narrow chimney (only 2 to 3 feet wide and 15 to 20 feet high) between the main wall and a detached block. Climb this chimney and a short 4th class pitch above it to the summit.
Because not many people are likely to have the GMS Journal I will also quote the essential parts of that description here, so people will have as much information as possible about the route.
"The route is quite obvious up to the "window" that frames views of the Livingston Range to the north. Above this window, the route travels up Class 3 cliffs to the ridge on the left. The ridge soon gives out against sheer cliffs. Everyone always looks around the cliffs to make sure the ridge does not continue, as the only possible alternative seems quite formidable. This "alternative" is a steep 20 ft. cliff (Class 4) that is easier to ascend than it appears. Above this pitch is a deep 15 ft. chimney that is usually ascended by walking to the far end of the chimney and stemming up to the top. Packs are usually hauled up in order to facilitate the stemming moves. Above the chimney is another steep little pitch that is climbed in the lower portion using opposition force. Gaining this obstacle leaves you only a few steps from the summit." Journal of the Glacier Mountaineering Society, 1988, p. 13, Route Notes, no author credited.
A rope is advisable.