It's unlikely that a school would extend the LSAT testing date requirement. I've been here on the forum for many years and have never heard of this happening. Applicants are able to write the LSAT multiple times within a five year period prior to applying, so there is ample opportunity to do so. I have heard of the application deadline being extended in certain circumstances but not the LSAT test date. You can inquire but it would be surprising if they agreed, especially given the increased number of applicants this year.
As for transfers, I wouldn't advise someone to ever accept a school with the intention/plan of transferring before they even arrive. Transfer admission is never guaranteed and you never know, year to year, how many spots will become available or how you will perform in 1L.
My advice would be to accept an offer from either Queen's or Western, should you receive one, and plan to be there for three years.