The wiring is fine, the problem is the spa draws 53-55A, you are supplying a 60A circuit. You will have the possibility of overloading the 60A breaker IF you are drawing max current from both the 15A GFCI and the 60A GFCI.
What I would do is run 1) 14GA THHN wire from the main panel as a 15A circuit, use the spa panel as a feed thru and then out to your 15A GFCI. You would need to run a neutral and a ground wire from the spa panel to the GFCI.
That's my CYA legal advice. Ignore anything after this.
As long as you are not going to have a heavy load on that outdoor GFCI, what you propose is not ideal but it will work. Just be aware that you do have the possibility to exceed the 60A breaker at the main and it will trip.
To be clear, 15A circuit requires 14GA wire.