If an LRT is built on Lakeshore you need to push the Hurontario Line underground and create an T for Lakeshore which is no big deal. The question then become how do you service both lines and interchange with the lines using 2 different gauges since Lakeshore will be an TTC line??
Mississauga Lakeshore line will likely be separate from the Toronto Lakeshore line. The benefit of running through service would be minimal, even if the gauge was same. Such a long line will be extremely hard to manage, and noone would travel end to end due to the trip time. Even today, few riders take the 501 all the way from Etobicoke to downtown. The 501's ridership in the west is mostly local, while the downtown-bound riders board further east, in places like Parkdale.
Mississauga Lakeshore line can just connect to the 501 at Long Branch, and those few riders who want to travel across the border can transfer there. In that case, no obstacles to using the standard gauge on both Hurontario LRT and Mississauga Lakeshore LRT.