I'm not very familiar with discordpy so I'll leave the actual review up to @TrizlyBear.
But I'm quite confused about this:

GuildCache[ctx.guild.id] = -1 is being used to identify guilds that are not part of the cache, before the bump they're assigned a value of -1 and upon api returning a 200, it is replaced with the current timestamp. Following that the bot relies on the built-in cooldown mapping instead of the api to handle cooldown.
In the scenario that bot restarted the cache would be empty, there remaining time before the next bump is obtained from the api response.
However a problem I encountered is the inability to overwrite the cooldown mapping to set the remaining time to a custom value (i.e time obtained by comparing with the timestamp from api response). Hence current implementation relies on the -1 flag to identify such cases.

Code looks good and is structured well.
Only thing I would suggest is moving the bot commands into a a sepperate cog file since you are using cogs for the error handler too. You don't have too but I'd say it's just a little more clean.
Overal everything looks pretty good and is well written.

LGTM!