It looks like the Phillips head bolt has a nut under. If so, you would back that off first, and the tool for that is a crowfoot wrench that mounts on the end of the shaft of a socket wrench:

Once that nut's a little loose, you should be able to back out the bolt, even though the head is a little wonky. As for the bolt that you broke the head off of, that will be tougher. But if there is a nut and you can loosen it with the wrench, you might be able to get a needle nose ViseGrip pliers on the bolt shaft and little-by-little unscrew it.

If you try to do anything with a Dremel, or with a basin wrench, you will likely make it worse.

And if the setup is different from what I've assumed, please explain and I'll see if I've got any other advice.

I think it must be the angle on the picture, but the Phillips head bolt is just the bolt to the larger ovalish shaped nut (brass colored). No nut in between. Technically, I guess it's bolt-ovalnut-washer.

One thing I'm not getting - I think the Phillips bolts are just putting pressure for the oval nut to the rusted washer that's above it. They don't actually go into any receiving cavity above. I don't recall having to align the screws to a hole - just hand tighten the oval nut to where ever it stops, then tighten the Philipps screw.

So I'm wondering, if I get the oval nut to turn, the Phillips bolts just come along for the ride, right? So it may not matter that I couldn't get them out first, no? And really I'm not planning to keep the faucet so if it damages it if there were receiving holes, so what? As long as it doesn't damage the sink basin (which is stainless steel with a lip and 4 holes)

btw, I ordered an extra large-jaw basin wrench. I believe it should work with the oval nut (one reviewer mention having a delta sink with the oval nut).

Cant see clearly, but I guess the Phillips thing screws into the brass thing and tightens it against the bottom of the sink. Loosen the screw/bolt and the ring should unscrew from the faucet shaft.

lilychen-

Ok. The photo wasn't great and I thought I saw a nut...forget my previous advice. What you're describing is a standard serup where you have two Phillips head bolts (not screws) which are applying pressure against the underside of the sink. You're correct that they don't go into holes. If you still can't unscrew the bolts a little you may still be able to rotate the oval nut with the basin wrench and unscrew the whole thing. If you can get it to turn, it should come loose after a rotation or two since those stuck bolts won't be applying pressure once you've backed off the oval nut a bit. This should work unless the bolts get caught on something as you turn the whole thing. You might need a second person to hold the faucet base so that it doesn't rotate while you're attempting to turn the oval nut.

Good luck, and please report back on how it goes.

Thanks kudzu9. That's what I'm thinking. I have the basin wrench on order and will go out to fix it Tuesday (it's at my rental house). Will let you know how it goes.

I have to take an angle grinder with a diamond blade to several faucets every year. Cut as close to the stone as possible without touching, and that recalcitrant faucet is coming out in minutes, I promise.

yup, this was the final attempt. The basin wrench was a tad

too large to fit in the space with the tub (do they have to texture it, or is that there just to grind off skin?) and the lip of the basin. And maybe because it was large but it was just really impossible to turn. So I had to hire a plumber and he sawed it off. i can do a fair amount myself but I definitely draw the line at power cutting tools in small or finished spaces ☹️ Shame they cant make these things more serviceable. Replacement was a basic glacier bay two handle. hope that will be less points of fault and easier/cheaper to fix in the future. Thanks for everyone's help.

lilychen-

Sorry it didn't work out, but appreciate the update. I was worried it might come down to that....