Offered a job, they are asking for a start date, my current company is in the middle of a project adn also pays a bonus in March...

I was/am considering a career change and brush up my resume and cover letter, I was made an offer and they are asking when I can start. I would be happy to work for this company and this is a role I am interested in, however I was not expecting a reply so soon... my current company is in the middle of a project that I am a part of and pays out its annual bonus in March and it is usually 10-14k USD (very significant amount for me), I want this new job but don't want to burn bridges by abandoning the project and don't want to forgo the bonus either.

How for can I reasonable push back the start date? Would it help if I told the offering company I would sign for the job to show them I'm not still shopping around for jobs/leading them on? Should I tell them the reason?

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