Separate the change from the project.
Figure out what the change will take separately. Then, you can either keep it as a separate set of tasks (preferred) or keep a before and after plan. Present your findings to the project sponsor, letting him/her know that the change adds to the budget and schedule. Get their buy in to proceed.
I've found that when presented with the impact of change, the project sponsor often has a change of heart (they are budget constrained as well) and agrees to postpone the work to phase 2.
As you move forward, if you have followed the preferred method of total separation, you will be able to track the project baseline and the change separately. The project actuals that you record will show the deviation from the baseline caused by the change and the change will have its own separate plan.