I bet the reason is explained ok already in other answers.
What to do:
You can make a bitmap copy with Edit > Make a Bitmap Copy.
Clip the problem area with the letters and make a bitmap copy of it. Place it onto cat. You do not lose anything, because the cat is already a bitmap. Everything else can stay as vectors.
The bitmap copy resolution (and bitmap export resolution) can be set in the preferences:
The resolution must be set high enough for your future printing plans, say 300 DPI.
Test also to trace the inverted area bitmaps back to vectors. The quality can be acceptable and the resulted vectors can be simple enough to not freeze Inkscape if their areas are reasonably small and the tracing is done with small enough color palette. It's tried here:
As you see, the traced inverted version isn't exact, stroke-like borders are definitely an unwanted result. But it can be acceptable for something.