What do you do about a giant gap under a bedroom door?

An inch and three quarters is past what any off-the-shelf sweep or door shoe is made to close. Most of those handle half an inch, maybe three quarters if they're adjustable. I ran into the same thing at my old place after the carpet got swapped out thinner, and the door was riding way high. What actually worked for me was screwing a strip of wood across the bottom of the door, then hanging a brush sweep off that strip. Painted to match, you barely notice it. On a stained door you can get the stain close enough that nobody clocks it. The screws go into the bottom rail, which is solid wood on most interior doors, so it holds fine. That got my daylight down to a normal sliver and it doesn't shift when the cat paws at it, which is the whole reason the rolled towel fails. Towels move. A screwed-on strip doesn't. One thing to check before you seal it up completly, though. A gap that big in an older house is sometimes deliberate, becuase the room has no return air path and the system needs air to escape under the door for the supply register to push air in. Close it off entirely and the bedroom can go stuffy, and the room goes pressure wierd with the door shut. If that happens, the fix is a jumper duct or transfer grille between the bedroom and hallway, and those come with blocking for light and noise if you shop around.

As for swapping the slab, it's not overkill so much as more work than people expect. You'd be cutting the new door to length, chopping the hinge mortises higher, and moving the strike. Very doable with a chisel and a Saturday, but the wood strip gets you the same result in an afternoon without pulling anything out of the frame.

For the 5am situation specifically, rigid beats soft every time. Once the door stopped moving when she smacked it, mine gave up on the whole routine within a couple of nights.

Not sold on the 'bottom rail is solid wood on most interior doors' part. A lot of the cheap hollow-core slabs in older houses have a rail that's barely half an inch thick, or in some cases just a cardboard spacer near the bottom. I screwed a weatherstrip holder to a closet door once and the screws basicly chewed through the veneer. Drill a small pilot hole and peek at what comes out before you count on that rail holding anything, especially with a cat thumping on the thing every morning.

This thread got me looking at my own bedroom door and now I'm wondering if mine was hung wrong from the start. The gap got bigger when we pulled out the old thick carpet and put down the wood floor, so wouldn't it make sense to just lower the door on its hinges a bit? Like unscrew the hinges and move them down the same amount as the gap? Seems easier than adding stuff to the bottom but I've never touched a hinge in my life, is that a bad idea?

Seconding the screwed-on strip idea. I did basically the same thing on a hallway door with a big gap and used a short length of aluminum U-channel instead of plain wood, so the brush insert slides in and I can pop it out later when the bristles wear down. Painted it flat black and against the dark carpet you genuinely have to look for it. Three years on and it has not budged, and the dog stopped nosing at the door within a week once it stopped moving. Wish I had done it years before messing around with the stick-on ones that just peeled off.

On the lowering-the-door idea, that works untill you look at the latch side. Drop the door an inch and three quarters and the knob's not lining up with the strike anymore, so you'd be patching and re-chiseling that too. Turns into a bigger job than it sounds.

Also worth checking if the gap's even across the whole width. Mine wasn't, and it turned out the bottom hinge screws had just worked loose over the years. Two minutes with a screwdriver closed up most of it on one side. If yours are sloppy from decades of slamming, tightening or swapping in longer screws into the jamb might buy you half an inch for free.

Update: tried the hinge thing someone brought up, popping the door off and dropping it down on the jamb side. Closed most of the gap, way easier than I expected. Cat still gets a claw under it at 5am though, so I want to add that U-channel strip with the brush insert before she figures out the new setup. How do you cut the channel to length, hacksaw okay?