I would urge some caution as to how sophisticated the Iron Throne’s bureaucracy is. As I’ve written about before, Westeros’ royal government is wildly uneven, with a relatively active fiscal bureaucracy but an almost non-existent judicial bureaucracy.

It’s also not clear how developed the financial bureaucracy is, because there’s a weird cross-fertilization between Littlefinger’s own criminal empire and the royal government. For example, wool and wine “factors” are not civil servants, but private traders who buy and sell on commission. Likewise, tax farmers are generally a sign of weak financial bureaucracies, as they’re essentially privatized tax collectors. 

Notably, most of the legit civil servants we see are related to ports: harbormasters, customs sergants, etc. This suggests a bureaucracy largely focused on enforcing and collecting tariffs and excise taxes, which allows you to collect taxes with a much smaller workforce, since you only need to cover ports rather than trying to collect taxes from vast swathes of rural territory. If we go by medieval England, the New Customs Act of 1275 required that each port had to have two collectors and one controller – as Westeros has five cities and about twelve or so towns which are clearly ports, I would guess that you’re talking about 50-100 men across the kingdom.