Is there some issue that I should be aware of with connecting the Arduino's ground to the welder's ground like this?
The solution is to power the Arduino from a supply that is not connected to mains ground. Examples include a laptop computer running from battery power, a battery, or a 2-pin wall wart (in UK/Ireland, that would be a wall wart with a plastic earth pin).
If such a solution was out of the question for some reason, then what would the approach be? Derive power and ground from the controlled device and use opto-isolated connections on the control lines? Something like that?