You really need two weeks for a round trip to the Rockies, 10 days if you fly one direction. With two weeks, you could do the Rockies, then fly to Vancouver Island.
However, accommodation will be a major issue in the Rockies, especially the holiday weekend of August 4-7. You need to book ASAP, and your itinerary may be dictated by availability of places to stay.
I'd make some changes. Skip Kamloops which has little to offer. Stay in Salmon Arm or Revelstoke for a couple of nights. Go to Banff or Lake Lousie or Canmore for a night, then up to Jasper, then back down to B/C/LL. You want to have two chances to see the Icefields Parkway - one of the most scenic drives in the world. Also doesn't make sense to split so many nights between Banff and Lake Louise, which are only 45 minutes apart. May also be a challenge to find something in one of the handful of hotels in Lake Louise at this point - and it won't be cheap. Then back across BC with time in Revy/Okanagan.
Jasper will be the hardest place for accommodations - get bookings there first, then adjust around those dates. You can do more or fewer days in Banff/LL/more before and after, depending on when you can book in Jasper.
That said, AirBnB and booking.com probably aren't the best sites to use -- with the high demand, fewer rooms will be listed there and with whole apartment/condo/house bookings illegal in the national parks, anything listed on AirBnB will be listed elsewhere.
For Jasper, I would try www.stayinjasper.com for Private Home Accommodations. They will likely have 2 or 3 night minimum stays, but are cheaper and less likely to be fully booked than hotels. For Banff and Lake Louise, all legal accommodations are listed at www.banfflakelouise.com Canmore is another option if Banff and Lake Louise are booked and/or too pricey. Note that the Hi Hostels in the area are very family friendly and many have family rooms. They can be a good, more affordable option.