EnrichR certainly looks rather nicely laid out and like it might be quite useful. GATHER doesn't look quite as fancy but also seems to provide some interesting information as well. Both seem like decent tools to add to what I've already tried, so thanks for the suggestions!
I think the main issue though is that without the background list, EnrichR in particular is returning very similar results to FUMA when conducted without a background list (not particularly surprising I suppose as they all rely on the same online resources to generate results). While useful, it doesn't solve the main issue that without the background list the results are a bit meaningless.
For example, both FUMA and EnrichR suggest that my subset of genes are enriched for immune response genes, but that is very plausibly because the original gene list was pretty enriched for genes related to immune response.