I think that's true - though with the caveat that there's a fair number of those people who are themselves live Greeks, Neapolitans and Bulgarians. As well as the US far right adoption of Byzantine imagery, there's definitely a streak of it in the southeastern European far right as well which I suspect does end up having some cross-fertilisation in various online spaces. I'm no expert on the current landscape of Greek fascism but given parts of the US far and religious right have had connections with the far right in Georgia, I'd be surprised if they didn't have similar connections in Bulgaria, Greece, etc too.
I think also the not wanting to see those people in their country isn't something they'd see as logically inconsistent - because they might very well say that what they want is for the Christian populations elsewhere to be ruling those countries. In that framework, they can justify refusing refuge because if someone is trying to flee from their proper place fighting for the reestablishment of an imagined Christian-fascist world order, that means they're a coward and can be disregarded. One saw a lot of this in wider popular discourses with Syrian refugees in Europe, the general sentiment of "well why don't they just stay and fight" was very common among a certain segment of the right.