Of course you can get out of Cat 5 on that bike. You simply need to finish ten races. You can also get out of Cat 4 on that bike, if you train and race well. And so on - it doesn't stop there. I started racing on a much older, much worse bike than that. You can race with practically any road bike, because the performance gains from a better bike are fairly marginal provided you are at least on equipment of reasonable quality and in good repair. If the hubs roll smoothly without binding and your shifting works reliably, the biggest difference will be in fitness and racecraft. The greatest gains to be made with equipment are in fit and handling.
But of course there are benefits to getting lighter, more aerodynamic and lower-resistance equipment. They are marginal, but they are real, and they add up. How much you choose to spend on those advantages isn't something other people can readily advise you on, as it will depend upon your overall financial situation, how much of a priority cycling is for you, OCD, and other things that you are in a better position to know than anyone else. The best we can do is tell you where the most bang-for-buck lies in terms for where you can improve your performance the most effectively. In no particular order, these are: bike fit, contact point comfort (shoes, pedals, saddle, handebars), tires and tubes. If you want to make your bike faster but don't want to spend lots of money, those are the first things you should be looking at.