I would say it depends. The question is why you are given that bonus. If it is for your good work in the year that past alone, then obviously you are entitled to it as it is part of your pay. But if the employer has stated that he gives the bonus to encourage you to put in good work in the upcoming year too, then it would be fair to deduct some of the bonus meant for that part of the deal.
You should check your contract, what it says about bonuses in that regard. From an employer perspective, I would suggest that close to every employer, at least in part pays out bonuses, not only to reward past performance but also to encourage future performance as well as to encourage loyalty.
That aside, I would not try to be sneaky here, if you go to your new employer and ask him to help you hide the fact that your leaving your current engagement so that the old one pays the out the bonus to you, the new employer just might change his mind about hiring you. I certainly would reconsider my offer to an employee that would do that as I would see it as a character flaw.
So, from a business perspective, don't gamble with the new job as it as you say pays considerably more. Be honest with both employers. Also, your former one might surprise you and pay you all of your bonus anyway. And if not, that bonus is a "one-time fix", the considerably higher pay on the other hand will come every month from now on. That’s probably worth much more over some time. Your current financial low you might instead take up a small loan for. Should that be necessary.
When it comes to morals, a good rule is, if we feel we might be doing something unethical, we usually are. It might not be in your particular case, but surely you are doing the right thing spending more time considering it.
In conclusion, regardless from what perspective you look at it, doing the right thing and being honest with will serve you best in the long run.
Good luck on your new job!