If it is a piece of paper credential that you want, my advice is to find an RYA-affiliated programme. ASA and other US programmes are well-accepted in the Caribbean and US charter markets, but the RYA accreditation is accepted world-wide. That's not a hard and fast rule, you can find Greek charter companies that will take ASA too for example, but the RYA goes a lot further afield. (N.B. - I learned to sail in the U.S. and have received most of my formal training under ASA auspices, despite sailing in the UK now.)
But Halmoore is right - your sailing resume counts for far more, and references help. And just having a solid experience base helps far more - maybe not in getting the charter, but actually enjoying it, especially when things go wrong, don't work, need repair, the anchor holding is poor, or the weather plays up.
I think the experience and tutelage you would receive from the PSC would be excellent. My only two concerns would be 1) do they have enough small boats (only 4?) to really get a lot of sail time on when you want to book them? and 2) if you are a civilian, how well will you integrate socially with the fraternal Navy members of the club?
I could add another depending upon your personal circumstances: I would guess that the PSC has a rather lower number of ladies sailing than civilian clubs tend to. That may not be a consideration for you, but I do remember fondly my days in the (strongly co-ed) Boston Sailing Club...(in point of comparison, the BSC had about 20+ small boats (Solings, Sonars, J/24s) to take out for personal use)
Where ever you decide, enjoy!