That looks like the one my son has. it's pretty tough. I don't think the 000-18 would do as good of a job as the Harbor Freight press. The wood is pretty thin.
This one is not that rigid. I don't think the 000-18 would do very good at it either...but I think I'm gonna just pull those nuts and bolts out of there, clamp it with a couple C-Clamps and weld it with cellulose rod. Otherwise it will take a bit of cleaning to be able to tig it. I only have tig/stick welders.