I used a similar swing (Boppy) which allowed for my son to be a little more reclined than the FP one you have. As long as his head isn't flopping around and he can't fall out of it, having him in the swing won't hurt him.
I can see your FP swing also has a neck support for infants, that's good - if it reclines more, then it'll be even better for him.
As a side note: My son was visiting his farmor today (grandmother) and she had the swing out for her other grandson to use... but Matthias got to it first (at 3 years old) and took a nap in it with it running.
LOL!