Find the py's for CLS to do that very thing, without stripping the rest of the army.
Your solution still does not fix the issue, as we would have one airborne, four light and five mechanized battalions. Is that really an ideal mix? To maintain an airborne capability for home defence, which was why our airborne forces were created post war in the first place, it means the unit could not be deployed outside Canada. So now we are back to square one, with one less mechanized battalion.
Another solution might be to concentrate all the LAVs in LAV regiments, and have them separate from the nine infantry battalions. I can think of a few reasons not to, including that seven troops are not enough to field a rifle section that is viable over time. When we went to war in 1939, in accordance with British practice, our sections were seven men. The experience on the BEF over the winter of 1939-1940 resulted in the restoration of the ten man section, even before the Germans attacked in May 1940.
It seems to me that we have wrestled with this bugbear more than a few times in the past. I don't think we have a solution in sight that won't lead to more problems.