Hi, everyone will say this but it's because it's true - Canada is huge and even with the internal flight you might want to be mindful of distances.
If you are flying into Vancouver, I would stay there for 2/3 days, you could possibly avoid the need for a car in the city to get to North Van and the bridge, there is public transport. The Grouse grind is fun if you are fit and ready for a challenge and have a baby backpack and the bears (they are unable to live in the wild so have a big caged area to walk around) and lumberjack show would be fun at that age up on Grouse Mountain. You can also bike around Stanley park (they rent baby trailers or baby bike seats) and the Aquarium at that age would be great.
As far as getting to Vancouver Island is concerned, you can go from the city from Horseshoe Bay to Nanaimo or from Tsawwassan on BC Ferries, try to make a reservation if you are taking a car. The journey is stunning. A seaplane might be a unique experience too, more affordable than you might think. Nanaimo to Victoria is an hour and a half. Nanaimo to Tofino is about 3 hours. The West coast is wild Pacific Ocean and is known for its storms. The East coast has a huge amount of tide going out, ideal for a little 2 year old to potter on beautiful extended calm beach. We stayed in Parksville in an apartment overlooking the sea, it had 2 bedrooms.
I would look at flying Victoria or Nanaimo to Calgary there are direct flights.
May in Calgary can still be cold, it is certainly unpredictable but Alberta is a bit like that. Prepare for all weathers including snow. Pay attention to the locals as well as regards bears, some of the best trails need at least 5 people or it isn't safe, restrictions can change too depending on sightings. Banff (or Canmore, it's a little cheaper) to Jasper is a classic trail for a reason, but you could also consider Glacier NP too.
One detour you might also enjoy is Drumheller Park which kids love, a place with tons of dinosaur exhibits and fossils.
You will need a carseat but I would suggest you rent one, they don't recognise EU/UK car seats and you may be fined if inspected by police.
Honestly with a kid that age and jet lag, I would try and make bases rather than continually road trip it moving from different hotel rooms each night. Parksville has aparthotels, Banff or Canmore or Jasper offering accommodations more like ski apartments where you can heat your own food rather than eating out all the time which can be tiring at 22 months.