Looks like I'm gonna have to get some smaller tools.
Looks like a 10mm or 3/8" tool. You'd have to go down to 1/4" (which are feeble little things) to fit that tool post. Generally, there's no disadvantage using a bigger tool, so if you can find a way of mounting them, use them.

If I were you, I'd have a close look at the tool-post mounting to see if you can reduce its height by 4-5mm (even if it means getting some material machined off the bottom), it'll give you greater flexibility in the choice of tool.

If you can find a cheap quick change tool post, it may allow you a greater range of vertical adjustment, plus you get the advantage of easy tool height setting FOC.