You should be able to use your AIP letter to get your Alberta Health Care. I don't think a registry office will accept it, but you can call Alberta Health and email/fax it in to them directly.
Below is a paragraph directly off the Alberta Health website:
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http://www.health.alberta.ca/AHCIP/temporary-residents.html
AHCIP eligibility while waiting for permanent residency status
In order to be eligible for AHCIP coverage you must have a valid Canada entry document or a letter from CIC indicating a “positive first decision” has been made regarding your application for permanent residency.
If you receive a “positive first decision” letter from CIC with respect to your permanent residency application, forward a copy of the document along with your completed application for AHCIP coverage and other required documentation to Alberta Health.
Your AHCIP coverage may be reinstated to the date of the “positive first decision” letter but not earlier.
If your Canada entry document is due to expire and you have not yet received this letter, you may be eligible for a one time only temporary extension of your AHCIP coverage. However, we recommend you apply for an extension of your prior permit during the waiting period to avoid a lapse of coverage.