I don't know - but even if I did, it would also be down to the law firm.
2 days is no time at all - paperwork could easily take a week or two.
You could apply now but personally I would wait until you have your paperwork.
Maybe email the Graduate Recruitment team and explain that you are keen to ensure you have enough time for onboarding for the LPC but also have some questions about how the LPC fee process would work and therefore can you arrange a call with them.
You could then get all your questions answered on that call (e.g. how long the paperwork would take and how the paying the LPC fees work, plus anything else you want to know). You can explain given the LPC deadlines you'd be keen to catch up this week if possible.
