A student in my course does well on exams, but doesn't do the homework: Go easy on them, or make them "pay the price?"

As an instructor, when I find myself in this situation, I invite the student to make an appointment to speak with me privately in my office.

During that private meeting, I will discuss his/her current performance in the class, and point out the likely consequences if he/she does not submit the required work. I will advise the student as to what he/she must do to achieve the grade he/she needs.

Then it's up to the student to earn the grade he/she wants - or to fail. You can't live their lives for them.

I strongly recommend you start with whatever the policy is in your syllabus. Most syllabi contain details about grading, points allocation, etc. By starting here, you can avoid any claims of "unique treatment", given that everyone received the same instructions.

That said, it sounds as though your student has a unique personal situation causing him to have difficulty completing all the material. In that case, I would follow the advice of ff524 and meet with the student privately to discuss. Simply bring the issue up and see what the student says. A comment of "I don't need to do your stupid homework, I'm good enough at the material without it" may deserve a different reaction than "I would love to do it but I simply don't have time with my other responsibilities".

Talk to the Student now

If possible you should communicate to the student exactly the precarious position they are in. Presumably this student is capable of completing the assignments and labs just like other students in the course. From the sounds of it, even doing an average to below-average job on these would substantially reduce the odds of a very negative outcome.

At that point the burden is on the student. You cannot make changes to the weighting of grades because of the external circumstances: this would really not be fair to other students. Presumably you have some notion of the percentage weights the various graded assignments are worth. Explain this, and make it clear that excellent performance on exams is simply not enough.

Of course, I would feel bad giving a D or lower to a student in this situation, but the grade must reflect the student's performance in the course as a whole. It is unethical to give a student a grade other than the one their coursework has earned.