Looks like a group stage is the way to go, despite Tharmi's protests.
Since 35 doesn't go into 4 any meaningful manner (a 6-team group stage is overkill, clearly), there should be a preliminary round with a plate tournament for the teams that are knocked out.
However, a 3-team plate seems ridiculous. Thus, I suggest the following:
Preliminary round:
22 teams
Group stage:
13 teams given seeds into group stage (based on CW BT Cup 6 performance, cup match duties, and lots if tied)
11 winning teams from preliminary round
That makes 6 groups of 4. Winners + 2 top runners-up go to quarter-finals. If so desired, can have a bowl and shield competition for the remaining 16 teams. If a seeded team is still BT-cup-tied that match counts as a draw towards the table. (This gives a team an opportunity to qualify even with 1 match off in the BT Cup, possibly even with 2 as attention will be paid to strength of teams when setting up fixture lists for these sides)
Plate tournament:
1 group of 6, 1 group of 5 (with option for a sixth team to join should anyone be interested). Winners meet in final.
This also means that the plate final and the cup final will be held on the same day. A team will play 6 games to win the cup, which is 1 more than would have been required with a straight knockout.