IANAL but if you don't object you still can't ask for a mistrial unless you argue that the defense given by the lawyers was literally incompetent, and I'm far from an expert but i think that's a pretty hard bar to reach. Especially if pre-trial they would put that lawyer on the stand and ask why they didn't object during the testimony but only complained after the testimony, and i can't imagine any valid argument that would be accepted by a court.
Bottom line, I'm not an expert at all, but if they purposely didn't object so that they could ask for a mistrial, I'm pretty sure that won't work at all