You've identified two separate issues that have separate answers. Increasing the load run through a load center is primarily an issue of electrical capacity, not physical space.
1. What is the wire size feeding this subpanel? What is the breaker amperage rating feeding this subpanel from your main panel?
2. What are you planning on running at the same time off of this panel?
Answering these two questions will tell you whether your current set up can handle the extra load of using a welder instead of a table saw.
Volts x amps= watts. Ohm's law. Let's say your subpanel is fed by a 100 amp breaker. 240v x 100 amps= 24,000 watts. This is the maximum amount of power that can be steadily run through this subpanel without tripping the main breaker. Now add up the wattage of everything you will run: welder, lights, heaters, etc. Are you under 24,000 watts or whatever your actual value is? If so, you can add the welder.
Now you can think about how to actually upgrade the circuit. Are you wanting to keep the table saw circuit, or change it from 20A to 40A? If you want to replace it, you'll need to remove the 12 gauge wire fed from the 20 amp breaker in your current table saw circuit, run new 8 gauge wire and feed it from a 40 amp breaker installed in the same two slots.
If you plan on adding a 40 amp circuit, you'll need to replace two of the 15 amp single pole breakers with high density breakers that have two breakers that fit in one slot and move the wiring. Now you'll have the physical space for a new double pole breaker. You'll have to check if your panel accepts this high density breaker though. It looks like you have a Siemens panel. They make dual circuit single pole breakers. I had them in my previous house.
