I assume you are going to BC? Their rules are as you mentioned, with coverage starting remainder of month you arrive + 2 more months.
However you will NOT get a waiver if you need medical help or deliver baby earlier. They specifically state pregnancy does not qualify:
https://www2.gov.bc.ca/gov/content/health/health-drug-coverage/msp/bc-residents/eligibility-and-enrolment/how-to-enrol/coverage-wait-period
Appealing the Wait Period
The wait period protects the province’s health care plans by reducing the likelihood of individuals coming to British Columbia for the purpose of receiving health care services at public expense.
While a process is in place to review appeals from persons who have received, or will require, health care services during the wait period, waivers are only approved in the most extenuating circumstances. Please note that this waiver process is intended for persons who have already established residence in B.C.
When an appeal for a waiver of the wait period is received, consideration is given to a number of factors such as whether a verifiable error was made by Health Insurance BC in assigning the effective date of MSP coverage, whether the person’s medical condition meets a requirement of having been diagnosed in the wait period, the seriousness of the condition, whether the person was able to access required health care services during the wait period and whether private insurance was obtained/obtainable for the wait period. In addition, the costs of health care received during the wait period must meet the requirement of being financially devastating. Routine pre-natal care and the routine birth of a baby do not generally qualify a person for a waiver.
So to continue with your plan, you need to consider a likely scenario you deliver early, or have other costs come up during pregnancy that will require out of pocket payments. I would budget around $10,000 or so to be safe, and of course best case scenario your baby can hold off coming out until just past March 1. Note though you may still require check-ups, ultrasounds, blood work etc during last 3 months of pregnancy which you will all need to pay out of pocket. If you can find a midwife to use, then your costs may be reduced.