I have not seen this issue in exactly this way on a Tempest. If it was water then it usually is a drop of water sitting on the lower face of the slot or hanging under the top but that is usually in lighter wind and in my experience is always a lingering bad reading and much stronger.
I am thinking that it might be an insect or something hanging on a spider web that blows into the gap for a moment.
But it can be a ‘glitch’. There are sometimes random readings which appear for unknown reasons. I once tested a Tempest for several days in constant wind inside and found occasional glitches can occur for no reason. Usually they would blend into the other values and not be noticed.
It may be a faulty Tempest, you could ask support what they think: Open a support ticket from top of this window to ask them.
In my personal display software I have included several algorithms which attempt to remove any glitches in any values from causing a disruption.
The historical link you gave shows the time of the gusts and their value on the graph but not in the table for the same time.
So if the graph used the same values from the table then the random gusts would not be displayed. That would be a weewx issue which I currently dont know.
And you say that the same gusts are also directly on the Tempest web site but I can not see that to know. Now as it is from several days ago it is not so simple to see the one minute values but they are still there. One way to get them is to use this google sheets method:
Not ideal, but another method to tidy the graphs may be to remove your gust line.
cheers Ian 