This is a classic project management challenge: fixed constraints (Time, Cost) with an undefined Scope. Since you cannot add more money or people, and the deadline is set, Scope is your only lever.
To maximize ROI, you must shift from "finishing a project" to "delivering the highest value increments" within the timebox. Since you cannot plan before Jan 1st, you must treat the planning as the first phase of the work itself.
Here are the specific strategies to manage this effectively:
Do not start "improving" anything on Day 1. Because no research has been done, you risk solving the wrong problems. Dedicate the first 5-7 days strictly to Discovery & Baselining.