Non-ASOPRS fellowships do exist, although I only know of a few. Pittsburg, UTSW (1 yr combined with neuro-op), Detroit Michigan (private practice), Coral Gables Florida (private practice), and Duke. I think there's a non-ASOPRS in NYC too, but not sure where. Moorfields in London also has plastics but obv it's in England. There is no centralized application system but rather by connections. The Florida private practice plastics was listed on SF Match in 2016 but not in 2017.
You can apply for ASOPRS status after doing non-ASOPRS training. It's not a guarantee but still worth a try.
One year non-ASOPRS spots may be valuable in that you "save" time by doing 1 year instead of the usual 2 years of training and still get to practice plastics. You can choose to do whatever you want after training--100% plastics or 0% or somewhere in between. It's up to you. I know someone who's doing 1 year plastics fellowship because they want to do another non-plastics fellowship afterwards.
If you want to get an academic position after training, however, university-based practices may prefer ASOPRS-trained plastics person over a non-ASOPRS trained person regardless of the volume of experience you got from fellowship.