Why not post a link to the item(s) in question - "steady stream of DC" is insufficient information.
Your LED accessories have internal electronics to convert the AC from a CW to DC as needed by the LED devices. Since the welder apparently needs DC, if you want to run it from CW AC, you need to perform this AC-to-DC conversion. With some soldering and component (diodes, capacitors, etc.) you can do this for a couple dollars. Or you can buy an AC-to-DC voltage converter module for, say, $5 with relatively simple 2-wire in, 2-wire out hookup.
Or you can use a DC-output adapter (a.k.a. wall-wart) like an unused cellphone charger.
More specifics on your options can follow after getting the DC requirements of your welder device.
