In my Super-Casual Pulp Campaign, I'm allowing players to choose the Mystic class from the Rules Cyclopedia, but with some modifications. Keep in mind that I'm using my B/X + RC Bridge Rules, and that I've also combined the Fighter and Thief into a single class, so when I refer to either of those classes below I'm really talking about both of them.
Saving Throws: Mystics use the normal saving throws from the Rules Cyclopedia (p. 30 and 109). Fighters use the saving throws from the B/X rules and the B/X + RC Bridge Rules. Thus, the Mystic and Fighter no longer share saving throws.
Treasure Donations and Oaths: On page 29, ignore the paragraph that begins "Mystics receive experience from [...]" and the paragraph that begins "A mystic's oath is his bond." On page 31, ignore the paragraph that begins "All the material goods [...]"
Smash, Parry, and Disarm: If these rules are not used for Fighters, they are not used for Mystics either. Otherwise, these rules work as normal.
Acrobatics: Ignore this whole section (p. 30-31). "Acrobatic mystics" are disallowed.
Thief Abilities: Mystics use the Thief Special Abilities Table from the Rules Cyclopedia (p. 22) as normal. Thieves use the Thieves' Abilities table(s) from the B/X rules and the B/X + RC Bridge Rules. Thus, the Mystic and Thief no longer share the same chances of success at these abilities.
Challenging Other Mystics to Gain Levels: To clarify, Mystics must challenge an appropriate opponent to one-on-one barehanded combat at every level above 9 in order to remain at that level. The Rules Cyclopedia could be read to imply that this only happens at level 10 (see page 31). Furthermore, the Mystic will be of the same level as their opponent during the duel, and only revert to the previous level if they lose. The Rules Cyclopedia could be read to imply that the Mystic does not gain the level at all until they win the duel (p. 31). If the Mystic does not immediately seek out their appropriate opponent to challenge upon reaching a level above 9, the DM may demote the Mystic to the previous level at their discretion.