If you regularly use a name other than your legal name, such that your work history is going to be under the "pseudonym", then there is no reason not to use your "pseudonym". If you want to submit a job application with a made-up name, there's no law against that either -- but it will create issues if the employer conducts any sort of background check, as nothing on your résumé will check out.
Obviously, if you lie on your résumé that can be deemed "résumé fraud", whether or not you use your actual name.
Your prospective employer does not need your permission to try to verify the claims you make on your résumé.

