In the BCA rule book, Section 1 contains the General Rules which include things like equipment specifications, how to spot balls, what to do if a ball "settles" and falls into a pocket, the definition of cue ball in hand, etc. Then the ruleset for each specific game begins with the statement:
Fouls are not described in the section containing the General Rules, instead fouls are described in a separate section, section 6, and the BCA rule book states that they are only fouls in a specific game "when included in the rules of the game being played." 8-ball, 9-ball, 10-ball, etc. present a list of the section 6 fouls that apply to those games (giving no description of the foul, just a reference to the section 6 subsection where the foul is described), and those games all include the cue on the table foul, while 1-pocket does not. In other words, the rules for each specific game DO NOT say, "All Section 6 fouls apply unless contradicted by the rules for this game." Rather Section 6 requires that each specific game present a list of the section 6 fouls that apply to that game. The ruleset for 1-pocket does not list ANY section 6 fouls that apply to 1-pocket, rather the 1-pocket ruleset describes fouls that do apply, and the cue stick on the table foul is not described anywhere in the 1-pocket ruleset.
According to he BCA rulebook, you can let go of your cue stick any time you want when you are playing 1-pocket. Therefore, if you are playing a tournament under the BCA rules, you can let go of your stick while playing 1-pocket, and it is not a foul.