As long as your K-1 status isn't within 14 days of expiring they should accept the application before or after you are married. If you can provide the required documentation it also shouldn't matter if you use the maiden or married name.
However, as others have said some offices might not have ever processed an SSN application for a K-1 and start coming up with goofy stuff like, "now that you are married, you need to apply for a K-3." The thing that throws some of them is the POMS procedure in RM 00203.500 says K-1 - fiancé of U.S. citizen and K-3 - spouse of U.S citizen so you are now married you are a spouse of U.S. citizen and must need a K-2 visa and an EAD card to be assigned an SSN.
If you want less hassles, apply in maiden name and don't say anything about being married. If you insist on going with the married name first or go back for the name change point this out to them:
RM 00203.210C. Procedure - Immigration Document as Evidence of Legal Name
When an alien applies for an SSN card, SSA presumes the name on the immigration document is the legal name unless the applicant presents evidence of a legal name change (e.g., marriage) that occurred after the immigration document was issued.
RM 00203.210 Changing Numident Name Data:
https://secure.ssa.gov/apps10/poms.nsf/lnx/0100203210RM 00203.735 E. PROCEDURE - ONLINE SAVE QUERY RESPONSE INFORMATION DOES NOT AGREE WITH IMMIGRATION DOCUMENT
1. Name
e. Applicant Has Changed His/Her Name But SAVE Query Response Shows Old Name
In some cases, the applicant may have changed his/her name after DHS issued the immigration document (e.g., he/she has married and is now using the spouse’s last name) but does not present an immigration document showing the new name. In these cases, the DHS system reflects the new name only when the person provided the name change information to DHS and requested to have his/her immigration record changed to show the new name. When the SAVE query response shows the old name that is shown on the immigration document presented and not the new name on the identity document presented, do not consider this a name discrepancy for enumeration purposes.
RM 00203.735 Requesting Online (Primary) Verification By SAVE:
http://web.archive.org/web/20051002025945/.../lnx/0100203735------------------------------------------------------------------------------------------------------------------------------------------
Plus this should throw them, tell I can change the name on my SSN record to my married name even if my status has expired, so why can't I do it now while my status is still valid.
RM 00203.510 D. PROCEDURE - SSN CARD REQUEST BY ALIEN NOT AUTHORIZED TO WORK
4. Correcting Numident Data When a Replacement Card Cannot Be Issued
If the alien does not provide evidence of work authorization or a valid nonwork reason, explain that no replacement card can be issued.
Although a replacement card cannot be issued, the Numident can be corrected if the alien has provided documentation supporting the change (e.g., a name change).
If the alien provides documentation supporting the change, process the SS-5 to update the Numident information but suppress the issuance of a card. (suppress issuance of the card, and see RM 00202.320 for providing SSN Verification Printouts.)
RM 00203.510 Alien without Work Authorization - Nonwork Need for an SSN:
https://secure.ssa.gov/apps10/poms.nsf/lnx/0100203510