My faucet valve cartridge is stuck. Need help!
As already suggested you're going to need a good penetrating oil with several applications over time.
As you've discovered the problem is going to be holding the valve in place while you try to get enough torque to loosen the cartridge. Typically most valves will have flat surfaces built into the valve body to enable you to stabilise the valve with a wrench while loosening the cartridge. In the pictures you provided I don't see an obvious place to hold the valve with a wrench. It appears that both the top and underneath mounting nuts rotate.
I believe your best recourse is to tighten the faucet onto the counter as firmly as possible using wrenches above and below the counter - which may require two people. You can then put one wrench on the mounting retainer above the counter applying clockwise pressure while you turn the cartridge with another wrench in a counter-clockwise direction.