I think you're getting too hung up on the word "university" where law schools say that applicants must have completed X years towards a degree from a university. A number of my law classmates had BBAs from a variety of institutions - a BBA from BCIT is the exact same as a BBA from Vancouver Island University and Kwantlen.
Granted, my opinion on the matter is pretty much irrelevant. If you want absolute certainty, call every school that you want to apply to and ask them if they have any issue with the fact that your 4-year bachelor's degree was conferred by an "Institute of Technology" instead of a "university".
I'm not sure if you've gotten far enough to narrow down potential law schools, but if you have, it would probably be worth it to contact them. LSAC likely won't have specifics on where your degree is acceptable. Best of luck.
No qualification is necessary to take the LSAT, although it's assumed that test-takers have at least two years of undergrad. I'm surprised BCIT wouldn't be on there. What kind of degree do you have? You should probably get in contact with LSAC.