Zack,
A change like this on a money term ranking page is always going to incur some risk. But here's what I think after your update.
/gifts/birthday-gifts ---> Better user engagement metrics, higher Page Authority, closer to the root.This curated landing page is best for a user who doesn't know which gift to buy, or even which type, but knows who they are buying it for a her or a him. I would say this describes an good (though it could be better) experience for someone searching for a short-tail, generic phrase like "Birthday Gifts" on Google.
/gifts/birthday-gifts/birthday-gifts
This landing page is more like a typical category page with filters and facets to narrow down the search by gender, price and other features.
They aren't the same page right now so I wouldn't use the Rel Canonical tag as a way to consolidate them. The View All category page is good for crawling, but the Curated Landing Page is better for users.
I would come up with a layout that combines the best of both pages and test that option against a percentage of the traffic to the /gifts/birthday-gifts. Basically create a B version of that page, which includes the filtering options available on the category page. These options can affect what shows up in the first carousel for "all birthday gifts" while the curated sections remain on the page for easy self-select (i.e. For Him / For Her).
Assuming that page converts at least as well as the existing A version, I would give serious thought to combining these two pages to see if you can get your highest performing landing page into the #1 spot. Given the relatively lower authority of the existing #1 spot, I think this is definitely doable.
You do risk losing some traffic that would have been more inclined to click on your listing after seeing more than one in the SERPs, but you can offset that with PPC, as you're doing now.
Collect your baselines first. Combine the totals from both pages to see if the consolidation results in more sales.