Yes, Oxford has it’s own aptitude test for PPE, whereas LSE only want your APs, PC, LOR, etc.
The TSA can only be taken in November of the year you apply. Your school can register to be a test center - it’s a little bit of hassle, so be sure that they will agree to do it, and that they have the registration completed by Oct 15 (which is also the deadline for the Oxford application).