You can book your flight and your cat's place on it separately, at separate times if need be. My flight was booked and bought by itself before my cat was booked by VIP Pet Transport, a service which is very affordable but only because they don't actually do all of the footwork, you still have to; they just tell you the steps and book the animal on your flight.
In fact my ticket was even booked and bought in the UK at a high street travel agents by my then partner, while my cat and I were still in the US. My then-partner sent me the date and flight details, I told all that to VIP Pet Transport, then they booked the cat onto the same flight. For some reason they said they had to do the pet booking, not me, but I don't know whether that was baloney or not and perhaps I could have just called the airline myself and booked him anyway, but whatever.
Taking your vet through the PETS scheme is manageable but it's a just freaking drag in my personal opinion. You can absolutely do it but it's all about details and, although I was amazed and relieved that in the final result I'd gotten everything right and the cat went through smoothly with not one thing wrong with the paperwork in the end, personally I'd rather have hot needles stuck into my eyes than deal with it again. I found the it stressful, confusing and a minefield of mis and even DIS-information, even the Defra site is crap because they omit to inform you of the minutiae details, although an actual phone call to them was perfectly helpful.
I could do it all again now knowing exactly what to do, but I hated it all and I'm glad it's over adn never coming up again. Meaning, sure it can be done and it's totally do-able by you yourself as long as you do your research, but its not fun, its confusing while you are learning your way through it all, and there are so many things you CAN get wrong that, even if in the end you didn't, it caused stress at the time of the doing of it, if you see what I mean. That was my experience anyway, for what it's worth. But by all means go ahead, it's stressful but CAN be gotten through.