You can disclose only the work experience you claim points for to avoid providing references for that.
But eventually you will need to mention everyone in your personal history.
And also you cannot apply for humanitarian and compassionate grounds if you had a negative decision from the IRB within the last 12 months. This is called the “one year bar.” (If the IRB decides your refugee claim is abandoned or withdrawn, that counts as a negative decision.)...
I would advise the sooner the better. Picking a right moment to come won't really help you. You will have enough hustle when you arrive, not only looking for a job.
To contact Immigration, Refugees and Citizenship Canada (IRCC) by phone from outside Canada, you'll need to use the international phone number: 1-613-944-4000.
It's not for a lawyer but a border officer to make that decision. The lawyer can overpromise you anything. What documents did you get when you left the border? Conditional removal order, BOC?