the transistors are similar to 2n2222 which have a peak 200ma base current
No they don't. It might be the peak collector current. Anyway that is not the issue you need to know how much current your relay takes. That is what actually flows. The transistor ratings are only how much then can take not what they will take. What they do take is determined by your relay and your base resistor.
The current available from the USB supply is negotiated in the software, 500 mA is the default value,
No this is the maximum value that can be supplied. No negotiations is going to be over that. Actually the default value is 100mA but the requested value is read from the EPROM inside the USB / Serial chip.
500ma is the max before a fuse blows in the usb
Only if you are lucky, some machines blow up the USB interface. (and it's not a fuse anyway it is a crowbar circuit)