Currently the Rockets don't have the cap room to sign Bosh to his max. If the Rockets don't intend to keep Parsons, it gets them a lot closer to having the room to do so. So waiting three days might make it easier for Houston to afford Bosh, while tying up Dallas' decision-making ability for the next day or two.
In short, I don't see Bosh forcing LeBron's hand, unless LeBron goes into next week, which I also don't see happening.