A rental seat is a seat with an unknown history. You have no idea if it's been in a crash, been washed with bleach between each use, NOT washed at all, who has eaten in it, peed in it, pooped in it, puked in it. You don't know that it's not expired or not recalled.
It's really not the best option. It shouldn't be an option at all.
If the Foonf is too heavy, how about a lighter weight seat for travel? The Cosco Scenera is cheaper than the Radian. Would that work for her age and weight and height? And a seat on the plane just makes sense. How else will she be safe on board? You're right that gate checking isn't an option. Unfortunately you don't have any safe options to choose from here.
A eat in a box that isn't super heavy can be gate checked, as long as you're ok with it being damaged or lost (though the box will protect it as best as it can. Most seats are shipped just fine that way, however, the baggage handlers are not shippers, they're trained to get things on FAST, not fragilely). That might be the safest of the unsafe alternatives. Just don't throw out the box because you'll need it coming home.
Much easier to just have a ticket for her, wheel her seat to the plane, put her in it, and then use it when you get there. Safer, too.
Wendy