There's this video that proposes how Light could have beaten Near. A critical element of the proposition is that Misa is accessible to Light, but was she? I seem to recall Misa was in the custody of the SPK up until some time just before Light et al are about to leave
She accessible, but not in the way the video assumes.
From the wikia episode plots:
Aizawa thinks to himself that the reason this is happening is so that Light can either be exposed or kill Near. As Light accepts, Near lays out some ground rules: all members of the Task Force must be present, as all members of the SPK and Mogi will be there while Misa will go elsewhere. Near then gives the meeting location: an abandoned warehouse he purchased known as the Yellow Box, and also forbids the Task Force from leaving the Death Note behind, thinking to himself that this is what Light would want.
The day finally arrives, and as Stephen Gevanni is on his way, Near announces that Misa has been moved to a luxury hotel, essentially freeing her. After Light confirms this, he asks Near to contact Mogi, who is with Lidner on their way, as to unlock the Death Note from a secure safe, each member has to put in a pin number that only each of them know individually.
Misa at that moment is a normal civilian girl, and there was no way for Light to safely get Teru Mikami (x-kira) contact Misa in the luxury hotel Near himself chose to placed her at whithout Near knowing. We can safely assume Near has a bug/tracker on Misa's cell phone (or can get a copy of the data). Or Mikami. Even if Misa is completely cleared, there is no way Mikami is not completely under electronic surveillance by 1.28.
Having Mikami contact Misa would have the same effect as not having his name written on the notepad. Light had very few pieces of DN left, if any by that time. It would be hard to find a piece where all those names could've been written (specially foreign names, with all those scribbly letters). And even if Misa was to text the picture of the names back to Light in 1 hour, it would give Near that much time to check the tracers and shut Misa down before she can forward the picture.
But the real problem was not assuming that Near would not find the DN inside a bank, the video gets that wrong. The unraveling of Light's plan was when
Mikami broke his character, took the initiative (instead of just doing mechanically what he had always done) and went to the bank out of his schedule to kill Kiyomi. Nobody expected Mikami, a creature of absolute habit, to do something so spontaneous. Had he kept to his schedule, Near would not have found the DN in the bank.
TL.DR: He could have done that, but it would give away proof of his alter ego as Kira to Near all the same.