How to negotiate a tenure-track assistant professor job offer in a very short time window?
While the decision to accept a tenure-track position is a big commitment, unless you have complicated requirements that were not discussed with the search committee/dean beforehand, it should be possible to make a decision on an offer within a few days.


  I want to seek any advice on the negotiation (e.g., start-up package,
  salary, starting date) if possible, given such a short time.


Think about your red lines and try to prioritize them: how much startup do you need, your preferred salary and starting date, teaching load, etc. and if the current offer doesn't meet these – you mention that there is not a lot of space for negotiation – you may prepare a counter-proposal asking for what you want but focus on the those points that are most important. If you cannot arrive at an offer that meets these requirements, politely decline the offer. 

Of course, if you surprise your negotiating partner (dean, head of dept., etc.) with a request that they weren't aware of, such as the need to renovate/construct lab space or a two-body situation where you're looking to secure a position for your partner as well, it might be impossible to have an answer from the prospective employer's side within a few days (some requests might need to be escalated to senior management or discussed with other departments/faculties). Your employer will tell you this, and if they really want you, they'll give themselves and you more time to complete the negotiation.

Don't expect to be able to re-open a negotiation after you have signed an offer. If you do have second thoughts, your only alternative might be to walk away. If you accepted the offer but then, on reflection, decided it was not for you and decided not to actually start the job, you wouldn't be the first (or the last) person to do this. Unless there are penalties in a contract (read the fine print), there's not much a future employer can do to you if you decide to re-neg on the contract. Be aware though that in some circles walking away from an accepted offer could damage your reputation – your future colleagues will know you did it – so I do not really recommend this course of action.

Finally, you mention that you are being offered a position that is not the one you applied for, so if you have a complicated ask (see above), one negotiating strategy could be to ask to be re-considered for the other position (if it has a longer timeline). Above all, be polite and positive, and keep the door open for future opportunities if you think this one is not going to work out and you like the employer.