File with the W-2 that matches the correct name and SSN. Keep all the documentation should IRS need to see copies of the W-2's you got. Also for your file, put down who you talked to at the company and that they refused to issue you a corrected W-2. If the return does reject in e-file go back to the employer.
If correct information was transferred in, he must have updated the file so everything should be fine.
You are correct a corrected W-2 should have been issued (W-2C).
Then in a couple months check with Social Security and review your earnings so you are getting credit for the correct Social Security paid in and the correct quarters.
https://www.ssa.gov/
