VUTURE

Hi VUTURE
This is likely due to the .eps file itself, many stock sites serve up '.eps' files which are actually .ai files in an 'eps wrapper'. This means that when importing these files into Affinity objects can be lost or rasterised unexpectedly due to the way the file is saved from Illustrator. 
Unfortunately as the .ai  file type is proprietary we can't read any information from it, meaning the imported eps usually has missing data - hidden where Affinity can't access it.
I hope this makes sense, apologies I can't offer a better solution here!

Hi VUTURE,
Here's the upload link. Thank you.

As you noticed boolean operations will not work for this since the resulting object must share the same fill/stroke attributes. I'm afraid there isn't much else I can suggest without checking the file myself and eventually pass it to the dev team to see how/what can be done in such cases.

Hi VUTURE,
Welcome to Affinity Forums
Disable all live adjustments and/or filters (in case you added any from Affinity Photo) that you are using (at least one is visible in your screenshot nested to a group) and enable them only when you want to export the file. Group vector objects that belong to the same subject/element and disable/hide the groups you are not working with at the moment. The less things Affinity has to draw on screen the better. Go to Affinity Designer Preferences, Performance section and set the View Quality dropdown to Nearest Neighbour (Fastest). Even so I doubt this will be enough for such a complex vector work. Seems you have vectorised/auto-traced a pencil drawing. Do you mind providing the file for us to take look please? I can provide an upload link if you wish to keep it private.