The TAGS Packet (0x5B) seems to be affecting the performance of players when they first render the texture of the achievement. Although this does not happen when someone gets a vanilla/common achievement (I suppose that is because it gets cached when the game starts).
I think the only time this occurs is when the achievement has a non-cached item. As seen in the video, the FPS-Spikes happen every time the TAGS packet gets sent.

This also affects all players connected to the server, even though the achievement is only being displayed to the player who clicks (in this case) the paper item.

Canceling the 'TAGS' packet to all the other players fixed the issue, but the player viewing the achievement will always have to deal with the FPS-Spikes.

The packet is this one: https://wiki.vg/Protocol#Tags