Welcome to CathInfo, Dan!
No, the SSPX doesn't "hate" the FSSP. They are only working for the triumph of the Catholic Church, and the universal kingship of Our Lord Jesus Christ. To hate anyone would be counterproductive to that goal.
You're especially welcome here, because we need some balance with so many sede's on here.
There ARE some non-sede members, but perhaps they are busier these days? Also some of them don't like arguing with the sedes.
At any rate, if you stay out of the "Crisis in the Church" forum you should be ok -- that is the only place on here where it's permitted to discuss the pope, the problems in the Church, etc.
I give people a decent amount of freedom on here, but I do have a few rules.
Again, welcome to the forum --
Matthew