Hi Ned,
Not sure what your experience level is with paint correction or if you have any equipment, but I have assumed none.
Whilst minor correction of your clear coat is perfectly possible as a DIY job, it should be noted that it is no quick undertaking to achieve good results. My biggest concern would be that you could get good results on the original paint (if the clear coat has not been corrected before) however, the paint defects on your spoiler may become even more obvious. Therefore, you might want to consider having a professional detailer assess the paintwork first, which should include measuring the paint thickness on each surface. That should give you a level of confidence on what can be achieved and the likely cost for a pro finish.
As a DIY job you will need at least one tool, plus several pads of at least two sizes, compound and polish, wax and some high quality microfiber towels. At the budget end on tools and good quality products, this lot will set you back at least £200-£250.
I am a big fan of this youtube channel, take a look at this video which has great info about how to choose a polisher. There are many more including how to undertake paint correction, choosing pads, compounds, etc, etc.
https://www.youtube.com/watch?v=wxtYZ9Pe2NA&t=710s
Hope that helps get you started!