I can tell you OP that in Ireland, Travellers 'white skin' doesn't confer a privilege.
Yes, even up against a person of colour.
One putative reason for this that discrimination against Travellers is ingrained in Irish society (not everyone but the casual expression of discrimination against Travellers, as illustrated in an earlier post, is still present & awful). However, largely Ireland was homogeneous in terms of skin colour until 20 or so years ago, when immigration meant that finally Ireland became more diverse.
There absolutely is racism in Ireland & we are very much at the starting point in terms of how to ensure equality for all communities.
But because we were at least a bit more aware of discrimination by the time Ireland became more diverse, in terms of colour, in your made-up line-up, a Traveller could very much be chosen & discriminated against over a person of colour.
I've said several times about Travellers poor outcomes still in terms of suicide rates for men, employment, life expectancy - these are all far below black people living in Ireland (please note: Ireland)
Please OP. Stop.
You are wrong about white privilege, in this context. I'm feeling quite upset, and I'm not a Traveller so I can't imagine what it would be like if I was.