Thought one. Using a 1 inch diamond core bit to drill around the anchor. This can work, but may be tight. While the bit is labeled 1 inch, its internal clearance might be closer to 7/8 inch depending on wall thickness. Since a 5/8 wedge anchor and sleeve can be about 7/8 inches in outside diameter, it might not fit. A better option would be a 1 and 1/8 or 1 and 1/4 inch core bit to make sure you fully cut around the anchor. Mark your depth on the bit so you don’t go too deep.
Thought two. Can you fill the oversized hole with epoxy and install a new anchor. Yes. If you drill out a 1 inch or 1 and 1/8 inch hole, you can still use epoxy to install a new insert. You can use your 5/8 inch female-threaded insert and center it carefully in the epoxy. Insert it slowly and rotate to avoid air pockets. Make sure the hole is fully cleaned and blow out dust before epoxy goes in. Even if the hole is oversized, the epoxy will bond extremely well. You’ll lose official certifications due to the oversized hole, but it will still be far stronger and safer than the spinning wedge anchor.
Thought three. Drill through the anchor using a rebar-rated bit. This is a destructive option but it works. An SDS-plus hammer drill with a rebar-rated bit can chew through the metal anchor. This will ruin the bit and possibly damage the hole, but once it’s out, you can clean it up and refill with epoxy and a new insert. Expect a rougher hole and some concrete blowout.
Another option is to cut the anchor flush with the floor using a grinder, then move the hole slightly and epoxy a new insert 3 inches or more away from the original. You can fill the old hole with epoxy grout. Not ideal, but workable if your anchor spacing allows it.
If you have a core drill, the cleanest option is to use a 1 and 1/8 or 1 and 1/4 inch bit to cut out the spinning anchor, clean the hole, and install a new insert with epoxy. If that’s not an option, drilling through the anchor with a hammer drill is your next best bet. Cutting flush and patching is the last resort.