Are there any known W[3] or W[3]-hard problems?
There are a few examples in the answer to Natural complete problems in higher levels of the W-hierarchy. In particular, the W[3]-complete problem $p$-HYPERGRAPH-(NON)-DOMINATING-SET may be useful for your proof.
Given that the linked question is now almost 6 years old, you'd perhaps expect some more examples to pop up by now. However, I couldn't find anything concrete in a quick literature search. The closest thing I could find was  Separating sets of strings by finding matching patterns is almost always hard (2017) by Lancia et al., where they conjecture that the problem they call PATTERN IDENTIFICATION is W[3]-complete for a particular parameter, but this conjecture is probably not very useful for your proof. I do note that I'm not an expert in this field, so there may be some more recent results out there I simply wasn't able to find.