Tape some thick cardboard down on the marble countertop around the faucet, then some plastic sheet. You want to protect the marble from impact and from the penetrating oil. Add a folded bath towel as extra protection when the hammer comes out (below).

Apply Kroil or PB Blaster, another penetrating oil product. Plenty of it. Wait a day, reapply, wait a day.

Get the correct six point closed end wrench. It needs to be exactly the right size to fit on the hex surfaces - too large and you will round off the part. This will be a large wrench, not a common size.

You can sometimes rent tools from auto parts stores. Set the adjustable wrench to fit the hex surface, tape the wrench jaw so it doesn’t move, bring that wrench to the store and ask for a six point wrench of that size.

While there, get a small tube of “anti-seize” and a small brass wire brush. I assume you have brass and copper polish, if not then get some.

Place the closed end of the wrench on the part, pressed all the way onto the hex surfaces, wrench handle more or less vertical. Again, the wrench must fit with minimal play; if you got the wrong size, stop and get the right size.

Have someone hold the head of the wrench firmly onto the part so it doesn’t slip. Take a large hammer and strike the wrench handle, at the far end. Start with little taps as practice controlling the hammer, then hit moderately hard. You are trying to use impact to break the thin corrosion layer that is holding the part stuck. The shock of impact works better than a steady pull. After the part starts moving, use the wrench in the normal manner.

The protection on the marble counter is there if you drop the hammer etc. The wrench is vertical so you aren’t swinging the hammer down at the counter. Be very careful all the same.

After you get the part off, and have replaced the cartridge, use the polish and a cloth to clean up the part and the rest of the faucet. Really polish it up well. The hex surfaces may be marred a bit (brass is a crappy material for a hex nut), but when polished shiny it won’t show (much).

Use the wire brush to clean up the threads. Apply anti-seize to the threads.

Reinstall the part, using only your fingers to turn it until the threads engage (because you do not want to use the wrench until you are sure you haven’t cross-threaded the part) and tighten it only just enough to not leak, then a tiny bit more.

Be gentle: with that big wrench you or more likely some ham-handed male helper can easily cross-thread the part or strip the threads.

This is how you’d break free then reinstall a large stuck nut on a car - e.g. an axle nut. Well, on the car you might use heat to break the corrosion, and you’d torque the heck out of the nut when reinstalling. But you’d clean the threads and use anti-seize, so that the next time isn’t such a nightmare.