I'm just wondering how you even filed the motion without a proof of service, but as long as you filed the motion within the 50 days, you will probably be okay. Here is a quote that you will find useful:
"`[t]he principal purpose of the requirement to file and serve a notice of motion a specified number of days before the hearing ([Code Civ. Proc.,] § 1005, subd. (b)) is to provide the opposing party adequate time to prepare an opposition. That purpose is served if the party appears at the hearing, opposes the motion on the merits, and was not prejudiced in preparing an opposition by the untimely notice.' (Arambula v. Union Carbide Corp. (2005),128 Cal.App.4th 333 [26 Cal.Rptr.3d 854].)" (In re Marriage of Falcone & Fyke (2008) 164 Cal.App.4th 814, 828-829.)
Reply to additional information: The most important thing is that you filed it on time and did your best to get it served. The clerk should not have accepted it without a proof of service being attached, but since it was accepted, you will probably be okay, since they will oppose it on the merits. As that case that I quoted stated, if they have enough time to oppose on the merits, then timely service of the motion is not an issue.
The way to proceed is see what they put in their opposition and what cases they use, rather than anticipate what they could use to oppose. Perhaps their stronger arguments will be other things. Besides, judges typically give people who represent themselves a lower standard to follow, but that does not mean they will let you walk all over them in court.