Unfortunately, Affinity Designer is not a program that I have, so I can’t tell you specifically how to pull this off via vectors. (In addition, there is something kind of weird that has to do with clipping paths and how it subtracts things in that program.)
One thing that you can do is to rasterize the image. That will convert it into exactly what you want to use for engraving.
Place the white text on top of the black circle, then rasterize it. (Turn it into a bitmap.)
I haven’t seen anything written up on how to do it in AD except to right click on a Layer and choose Rasterize. If the text and circle are on different layers, you might not be able to do that, since they need to be combined.
The other option is to take a screen shot of the image and save that as a jpg or png file. It’s a quick way to rasterize. Then you can import the jpeg into AD and draw a vector cut line around it in the AD software. Anything pure white will not be engraved, anything black will be engraved.
There are a couple of folks here who use AD that might be able to give you some other suggestions… @tim1724 I think, and @Xabbess.