Why the "first" Enterprise would disappear from Christopher's Mk 1 Eyeball Scan and radar alike is not all that difficult to explain. The ship was about to make a recovery in any case, and would have ascended away from the interceptor and perhaps engaged radar-thwarting shields mere dozens of seconds after the moment where the "first" Kirk made the decision to use the tractor beam instead.
The "second" Enterprise confusing Christopher with the beam-superpositioning at that crucial moment (from the secure heights of outer space) would buy the "first" one those seconds, allowing it to ascend without ever engaging the tractor beam, and that would be that. No real chance of the messed-up "first" ship spotting the "second", either, before the "second" one proceeded to the future*.
It's too bad that they did the beam-merging trick in the exact wrong order, though. They were coming in from the future: surely it would have been natural to beam down the Sergeant first? And beaming him down into himself, aka confusing the "first" Sergeant at a crucial moment so that he wouldn't notice the "first" heroes stealing the radar tapes, would nicely erase those tapes from the resulting timeline. Now they stop Christopher from spotting the ship first, but the tapes are still there - and now the "first" heroes never beam down to recover the tapes!
(Or perhaps they still do, after reaching the safety or orbit? But they would do it in a slightly altered fashion now that they weren't caught by Christopher, and thus Spock couldn't really calculate the right moment to mess with the Sarge, not in advance anyway. But he does take his sweet time, potentially to scan. Perhaps he plays chess with live pieces there, observing brief glimpses of timelines and calculating the cascades of events, then intervening?)
Tmo Saloniemi
* ...which now might enjoy the presence of two Enterprises. A single-timeline, BTTF style model works there, after all, but it would lack a mechanism for erasing either of the two ships after all was said and done.