Can you avoid your promise to arbitrate a dispute, or slow the process down, if you and the bad guy also have a lawsuit going on at the same time?

Nope. Promega Corp. v. Life Technologies Corp., No. 11-1263, slip op. 12 (Fed. Cir. Mar. 28, 2012) ("The district court's duty to compel arbitration is not altered by the fact that non-arbitrable claims may remain pending in the district court.").