Do you think I should remove all the low lying/horizontal branches near the base as well?
On my allotment site, fruit that low would be eaten by muntjac deer so I would take off branches like that, but even if that is not a concern for you, I think I would still lose them to make it easier to get to the centre of the tree to prune it.
The split trunk should be fine, but you might want to make sure you thin out branches that point into the centre of the tree and towards each other, just to make sure all the fruit gets enough sun.
My rule with tree pruning is do a bit and then spend the next couple of plot visits having a look and a serious think about whether you need to do more. It is easy enough to decide to have another go - less so if you are trying to stick a branch back on because you regret cutting it