The OP does not want to do a name change from the courts. She wants her birth certificate to stay the same as it is now.
She wants to assume her married name, her province will allow the married name on her new documents just not the placement of her maiden name as a middle name.
Yes it will be easier to assume her married name and get that started with her name change on the passport. That way the visa and GC will not have to be changed later and save a bunch of money.
Also nothing in Canada's system is fast getting a court date can take months and years to happen depending on where she lives. Second ordering a new birth certificate with the name change will take months. Then applying for the new passport.
If the OP cannot amend the new name on the marriage certificate, I wouldn't worry about it. Most government forms and ID will often not even list the middle names, never mind two of them.