Yes that would be an issue. Is it possible to update your SSN at your current place.
I've been working at my current job for nearly 7yrs w/o SSN. So you kind of bring up a good point about the background check. Because if I don't put my current job, than it would be like I had to no experience and had been sitting at home for the last 4yrs since graduation.
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DACA Approval: 3-19-2013
DACA Renewal 1 Approved: 3/19/2015
DACA Renewal 1 Approved: 2/09/2017
"Patience is the key to Success" - Bill Gates