It all depends on whether you're willing to lose the job offer to keep the bonus. If so, just tell the company making the offer that you won't be able to start

It all depends on whether you're willing to lose the job offer to keep the bonus. If so, just tell the company making the offer that you won't be able to start until whatever date you want to say. If they say no, try countering with a signing bonus for the amount you're losing out on - or some percentage of the amount you're missing out on. Whatever you're willing to accept. If you're not willing to lose the job offer, just accept it and give up the bonus.