I am not sure I would beat the k... out of my glider. They make a
tool called a J-Roller for this purpose. Not real expensive, and
might be hanging right next to the contact cement at your local home
center. Or find a friend who likes to make cabinets (Formica type
stuff) and borrow his. Then roll the k... out of everything.

I agree, wait till a warm day. If it is too cold, especially for the
non-solvent contact cements, they take forever to dry to the tacky
state.

And don't forget, once the 2 pieces touch each other, they are not
going anyplace. Given all the weird shapes you will be working with,
I would figure out the final assembly BEFORE any cement is applied.
Wax paper, parchment paper between the glider and carpet might work
well to permit you to reposition before you slowly pull the paper out.