|Agree with Linda and Clem on this.|
I'm in California. In my state, the signer can sign the way indicated by lender/title to sign, how it appears under the signature line, i.e. name aka alias. However, the notarial certificate is strictly the notary's purview* and unless that signer has unexpired government-issued photo ID in both names, the only name in my notarial cert will be the name that appears on the ID presented to me. So, if that notarial cert is pre-printed, I'll either cross out the non-ID name and initial or I'll attach a California-compliant certificate with the appropriate name. (*In California, certificate is to have specific verbiage; venue/date/name to be supplied by notary.)
I've done this many, many times without any issue. The first time with one particular TC, I had to state my position and send them the excerpts from the California Notary Handbook corroborating my stance. After that, they usually call me primarily for their high-end clients.
It's okay to push back if you know you've been given instructions conflicting with your notary laws. But, you must be able to prove your point. You'll be respected for it.