William Baffin? William Baffin is not a species rose. You might mean that it was own-root. Own root roses can have suckers, and many do. That sucker is also William Baffin. You can try to carefully sever it from the "mother" plant and plant it somewhere else.

Any plants that you might get from seeds of William Baffin might look the same, but wouldn't be William Baffin.

very unlikely this is a seedling

I understand when you define own root rose. Can you please define what a species rose is? Will rose seed from a grafted rose have seed of the blooming rose or of the grafted rose? If a grafted rose has suckers will it be of the orginal root stock and not the blooming rose? Thanks.

Species roses are the origin roses, the earliest. You will see them referred to as "Rosa something", for example Rosa multiflora (also noted as R. multiflora) is a species rose. See the link below for a better description. Most of what we grow now are hybrids.

Seeds will be from the variety that is grafted onto the rootstock.

Suckers could be from the rootstock or from the variety. If the sucker comes from below the graft union, then it is a sucker from the rootstock. If the sucker comes from above the graft union, this is new growth of the grafted variety, we usually call these "basal breaks".

Here is a link that might be useful: Old Garden Rose Primer by Steve Jones