I am in a very similar position to you! My PR card expired over 5 years ago.
I plan to go to Canada soon. On arrival, I expect to be reported for not meeting the Residency Obligation ("RO") under s.44 report. As far as I know the immigration officers cannot stop a Permanent Resident from entering Canada when they are in breach of the RO, but they may report them for the breach and give them a piece of paper that confirms that.
I then plan to appeal the Removal Order they will give me at the airport by filling in the relevant form available from http://www.irb-cisr.gc.ca/Eng/res/form/Pages/IadSaiAppForm.aspx (you can choose which city you want to appeal in, Vancouver, Toronto or Montreal) and submitting within 15 days (I think 30 days is the limit, but getting it in on time is important). The form is not difficult.
Then I plan to apply for a renewal of my expired PR card and obtain a temporary 1 year PR card. The renewal form is available here: http://www.cic.gc.ca/english/information/applications/prcard.asp
In case I do get reported for not meeting the RO, as far as I know I am allowed to apply for and obtain a temporary 1 year PR card while waiting for the appeal date at the IAD. If the hearing takes longer than a year, then you are allowed to keep applying to renew the 1 year PR card when it gets close to expiry.
Yes you can work while you are on appeal. You need an SIN to work. If you don't have one already, you need a valid PR card to get one, which you have. What I think you may have difficulty with is to sponsor your child. I am not expert on that aspect...
I am hearing that in Vancouver it takes about 18 months to get in front of the IAD but in Toronto, as long as 2 years or more in some case. I noticed some cases on the http://www.canlii.org/ website for 2016 that some cases for RO appeal are taking more than 2 years to get a date to present your appeal at the IAD (see http://www.canlii.org/en/ca/irb/nav/date/2016_03.html). Here is a specific example: http://www.canlii.org/en/ca/irb/doc/2016/2016canlii21938/2016canlii21938.html
My strategy, therefore, is to spend up to 2 years working in Canada after being reported for not meeting RO. Then, in case I lose my appeal, at the IAD on H&C grounds, I will simply get my employer to give me a high-skilled permanent job offer so that I can be in priority position to re-apply for PR based on express entry in the future.