The most important thing to do is set clear goalposts. What criteria do they actually require in order to take something seriously? If they give you clear criteria, it should not be hard at all for you to meet it, and if they go back on what they said, that is an opportunity to discuss their mindset, since I agree with you that they probably are just looking for excuses, but fundamentally that is something between a heuristic and a vibe and we don’t have firm enough evidence to be able to make a claim like that and be taken seriously by people who aren’t already sympathetic. If you patiently discuss with them what their supposed criteria is, meet that criteria, and then they make a new excuse, you are much more obviously justified (though you shouldn’t take it as just an excuse to disparage them, but rather an opportunity to ask them about their mindset and engaging in good faith).

Also, as others said, I’m pretty sure it is usually printed with lots of footnotes for sources, having a literal bibliography is just a formatting thing (even if it’s a better practice than not having one). I need to wonder if they aren’t either parroting a gotcha or just flipped to the back in hopes of finding that there wasn’t one. But again, even of there’s a real likelihood of that, it doesn’t mean that’s a good reason to start slinging accusations. Instead, you can just point out that there are indeed sources (you can check with them first and see)