How to approach negotiating the start date of a tenure-track position?
Let me offer the perspective of a former department chair who was involved in hiring negotiations from the department side. Here are a few things to keep in mind:


Hiring a faculty member is a long-term investment. When a department recruits a new faculty member, typically they expect the person to be around for the next 20-30. When viewed through that prism, any short-term concessions, especially ones that come without any financial cost like a deferral or unpaid leave of absence, are fairly negligible compared to the total benefit the department expects to get out of an otherwise-successful recruitment.
Leverage is important, but it’s not everything. An academic recruitment is not usually a zero-sum game where each side tries to exploit the other to the maximal extent they can get away with. Specifically, it is a repeated game where the reputation of a department’s recruitment practices matters a lot for its ability to recruit successfully in the future, and besides, the two parties are going to be working alongside each other for a long time, so it is in everybody’s interest that things are done amicably and in a way that leaves the other side generally happy and satisfied. For this reason, a candidate’s request that can be granted by the department for a relatively low cost will typically be looked at pretty favorably, even in the absence of leverage. (At least this is true in departments and universities run by rational, competent people...)
Keep your cool. From the language of your question it seems to me that you are approaching the discussion from a rather emotional place. For example, you say “I hate to do this”, suggesting that you feel guilty or apologetic about bringing up what you perceive to be a super-delicate or taboo topic. Relax! Such discussions are all in a day’s work for the people you’ll be talking to, and don’t carry nearly the emotional weight that you imagine them to carry. Nor do you have anything to be apologetic about. “Negotiation” in the current context is just a code word for you telling the department what your needs and preferences are, to which the department will respond by telling you what its needs and preferences are, and by a process of joint exploration you and your co-negotiator will arrive at the point in the space of possibilities that is reasonably agreeable to you both. Just bring up the topic of the deferral and anything else that’s on your mind, in a live conversation (ask for a phone/video call with the department chair), and do so in a straightforward, mature, honest and professional way, and you’ll be fine. The worst that can conceivably happen is they’ll decline your request and you’ll be no worse off than you are now. 
You should bring up your request as soon as possible. Given the short deadline, any further delay weakens your position, since if you wait much longer, people will wonder if things that you say are important to you maybe aren’t as important as you say they are (otherwise why are you only now remembering to bring them up?). By acting fast you will appear more credible and your requests are likely to be taken more seriously. There will also be more time to explore possible creative solutions that will satisfy both you and the department.


Good luck, and congratulations on the offer, which I’m sure you worked extremely hard to get and is well-deserved.