My worker difficulty seems to be stuck at around 200, even though I set the minimum difficulty much higher on the worker settings page.
Do you mean it is stuck at exactly 256 all the time?
Bitminter worker difficulties are powers of 2, with a current minimum of 64. So 64, 128, 256, 512, 1024, and so on. If you don't set a minimum then you will be started on 512 currently.
The mining server will adjust up your difficulty if it makes sense for your hashrate. The aim is to see roughly 20 work results per minute from your worker. You can change this number in the worker settings. You can also set a difficulty minimum which is mostly useful to make the initial difficulty ramp-up faster when you connect to the mining server to start mining.
Note that from you change difficulty settings on the website until you see it active on your workers can take up to 20 minutes (worst case).
Mostly you shouldn't need to set minimum difficulty or the work results per minute target. The mining server will start you at 512 and adjust your difficulty as necessary when it sees how fast you are mining.