Your welding Lead acid battery, or the connections from it to and through your welder, potentially not having enough juice, or allowing enough juice to pass, has nothing to do with my spot welder.
I jumped in only as so many people seem to think old lead acid batteries can be magically restored after being abused.
Abused for lead acid batteries is a dependent on how they were treated.
They ideally prefer a prompt and true recharge to 100% State of charge as often as possible any time they are discharged.
Sitting at 80% charged and less ages them faster.
it takes no less than 3.5 hours to recharge from 80% to 100% state of charge, with lead acid, and only when the battery is still healthy and when it is held in the mid 14v range for that duration. Less healthy lead acids will need slightly higher voltages held longer and sometimes ridiculously longer.
A healthy 800CCA battery should have more than enough for a spot welder.
An unhealthy one, even held at 15.7v via on a 50 amp potential power supply, likely cannot.
I Lived on lead acid batteries for 20 years. I learned how to get good to excellent service life from them in deep cycle duty, and also as starting batteries.
They are finicky in regards to their treatment, and build quality is a huge unknown, as as soon as one brand gains a reputation for longevity, maximum executive profit is not as high as they seem to think they deserve, and can get away with, so corners are cut, quality control reduced, and the battery is designed to just barely last the warranty period.