tl;dr: arm swing is noise compared to accuracy of GPS
Is there a GPS watch that somehow accounts for this to produce more accurate data?
No, there is not.
The reason is among others that to get that level of accuracy for GPS requires significantly more than the GPS locator can provide.
The U.S. government is committed to providing GPS to the civilian community at the performance levels specified in the GPS Standard Positioning Service (SPS) Performance Standard. For example, the GPS signal in space will provide a "worst case" pseudorange accuracy of 7.8 meters at a 95% confidence level. (This is not the same as user accuracy; pseudorange is the distance from a GPS satellite to a receiver.)
It is possible the military, who are allowed multiple GPS channels could account for the noise associated with civilian GPS and create a watch this accurate. They may in fact have one. Probably not one you can purchase, however..
Note that for applications that do need this level of accuracy such as farming or surveying all have some level of supplemental signal (such as RTK). These tools are all several orders of magnitude more than your $90 requirement - most are $1000-$10000 price range. They are also bulky and not likely to be portable. Many still have accuracy that is only in the ~10 inch range, too. If your arm swing is 2 feet their error is still nearly 50% of what you are wanting to account for.
Also observe that not all running watches are created equally. I would suggest looking at this and picking one which has the best criteria.
Now, onto my next point, which is instantaneous pace.
The pace should be accurate to the second - I've noticed that my current watch only does this in multiples of five seconds (e.g. it will rounds 7:03 up to 7:05).
First, keep in mind how pace is calculated.
It is almost assuredly not polling the GPS signal multiple times every second and calculating instantaneous updates every quarter second (or whatever). Because GPS is a battery hog to get lightweight devices it likely is polling at an interval. A friend of mine with a PhD in networking technology indicated that once/second with a GPS intensive app is probably a good rule of thumb. More kills battery life.
So lets say you are running a 7 minute mile. That's about 10 feet per second. Let's generously say that the accuracy of the GPS error is only +/- 3 meters (remember, 7.9 meters is the published accuracy at 95% confidence). This is a pretty significant improvement - a 7.9 meter circle vs a 3 meter circle. 3 meters is about 10 feet, and so this means that any calculation looks like:
- 10 real feet traveled +/- 10 feet = 0-20 feet GPS traveled, in 1 second
So clearly a 1-second pace doesn't make sense from a GPS perspective. Let's look at 5 feet:
- 50 real feet traveled +/- 10 feet = 40-60 feet GPS traveled
Ok, this is better, but still not super accurate, even with a 5 second polling interval. Keep in mind this is assuming that the GPS accuracy is pretty good that meaningful and repeatable.
Let's also consider your arm swing. At worst, your arm swing might be 2 feet linear distance - out of 40-50 feet traveled in the 5 second interval. This is a small error margin, perhaps 2/40 or about 5% maximum error. But the GPS error is still 25% of this scenario.
If we consider the 1 second poll, the GPS error is 100% and arm swing is still only 25%.