This one is a bit tricky for me. You should definitely apply for a study permit extension, as your current permit expires before you will finish your program.
However, I doubt you can continue working, even part-time until your final semester starts. You mentioned the school requested that you complete the placement term starting January, so I assume that would mean you defer your studies until then so this would mean that you can't continue working. Maybe someone more experienced can provide further insights into this.
see below excerpt, parts highlighted in
red :
What counts as an authorized leave from your studies?
There are a few cases where you may be able to take a leave of up to a
maximum of 150 days from your program of studies and still be considered to be actively pursuing your studies.
You don’t need to tell us if you’re taking an authorized leave. However, if we ask for it, you must provide proof that your leave is
- authorized by your DLI and
- no longer than 150 days
It counts as authorized leave if
- your school has authorized a leave from your study program for
- medical reasons or pregnancy
- family emergency
- death or serious illness of a family member
- any other type of leave your school authorizes
- your school has closed permanently or because of a strike
- you’ve changed schools
- you or your school deferred your program start date
- In this case, you must start your studies the next semester, even if it starts sooner than 150 days, and get an updated letter of acceptance.
You can’t work on or off campus during an authorized leave from your study program, even if your study permit says you’re allowed to work in Canada.
https://www.canada.ca/en/immigration-refugees-citizenship/services/study-canada/study-permit/while-you-study/study-permit-conditions.html