Okay, this turned out to be quite a long answer describing a few arguments I've com up with. The short answer is: maybe, maybe not, but I think it's possible. There are arguments to be made for either side of this discussion, and I don't think you you can ever be absolutely sure unless you find an expert on both astronomy and Chinese folk tales (which I don't claim to be), or invent a time machine to go back 2600 years and check for yourself.
The most obvious argument to be made in favor of the hypothesis that these magpies are the Perseid meteor shower, is that folk tales referencing stars are very likely to be based on astronomical occurrences during that period (such as a meteor shower, perhaps).
The (in my mind) most obvious argument against your hypothesis is what you suggest yourself: Perseus is quite far away from Vega and Altair. However, Perseus is the radiant of the shower, which means that all perseids will appear to come out of Perseus. Because of this, it is perfectly possible for a lot of perseids to appear between Altair and Vega, parallel to the Milky Way. It is, however, impossible for them to go from Vega to Altair, or vice versa, since they wouldn't come out of Perseus anymore.
You suggest that this could be explained by the precession of the equinox, but a quick lookup on Stellarium (which is a great (and free) tool if you're into astronomy) tells me that even 2600 years ago Perseus was quite far away from Vega and Altair (actually, that's quite obvious, since the precession of the equinox causes stars to move relative to the extension of the north pole, but not relative to each other, as far as I know).
Another argument to be made against this hypothesis is that the perseids are by far not spectacular enough to be seen as a bridge over the Milky Way, even when they all appear in the required spot. During the yearly peak of the perseids, you are expected to see about 1-2 meteors per minute, which can be noticeable at best. This argument is then countered by the fact that the ZHR (how much meteor you see per hour) is not the same every year, but can peak on years when Earth passes through a fresh dust cloud left behind by the comet that causes the shower (which, in this case, is the comet Swift-Tuttle, if you're interested). Therefore, it is perfectly possible that the perseid meteor shower was extremely active 2600 years ago, causing the creation of this story. Such peaks are somewhat rare, but not too rare for this hypothesis to actually make sense (in fact, we will be seeing such a peak in 2028).
The fact that the dates of the festival and the peak of the perseids are a few days apart and not exactly the same is not quite an argument against your hypothesis, since it is perfectly possible for the date of the festival to have moved slightly due to differences between the (old) Chinese and Western calendars. I'm not an expert on calendars, however, so I can't say that for sure.
Given all these uncertainties, your hypothesis (along with what you mentioned yourself in your own comments) seems plausible, apart from one thing: we don't know whether Swift-Tuttle was already active 2600 years ago. If it wasn't, the perseid meteor shower obviously did not exist (note that comets don't exist forever: some gravitational variation causes them to exit the Oort cloud and enter the solar system). We don't know when exactly Swift-Tuttle started being a comet.
That is only a weak argument, however, and when it comes to astromical hypotheses, the most convincing argument is usually a principle which is referred to as Ockham's razor: if given multiple explanations, choose the one that requires the least amount of assumptions (or the most plausible ones).
Personally, since the date of creation (2600 years ago) is very vague (the uncertainty on this must be at least a hundred years), I think it is perfectly possible to assume that the perseids peaked somewhere during that period, causing the creation of this story.