LED Driver with relatively high capacitance on PWM input causes PWM signal to be laggy
Design Specs


As your TV and computer monitor only have 8 bit intensity control per channel, I think your 16 bit control spec is a overly optimistic for optics.  
I suggest 10 bit or 0.1% is adequate  ~ 60 dB range and 12 bit is presuming too much.
1% is often adequate for lighting control and 1% of  1kHz PWM max = 10 us although they do not indicate the resolution limit but this matches your present result width .


Calc. Errors


The Thevenin equiv. R  of 22k/3.9K is not 22k but rather 3.3k for a logic "1" which suggests a much higher Cin if  Cin=T/R and T is measured for a 0 to 64%  asymptote.


Suggestions:


Consider  open bin Rank (top row on spec p4) 2.5~30 mA.out / 5mA.in = equiv to hFE= 0.5 to 6 (CTR)  
This if Vf= 1.2 and AVR=5V+50 ohm then If = (5-1.2)V/(220+50)Ω = 14mA so worst case CTR=50% = 7mA out.    
I am assuming you got open rank(bin)  : Verify
PWM spec = : DIM ~ -Vin >2.6 ~ 5.5VDC or open circuit
therefore the Vin threshold is ~ 50% of 2.6V = 1.3V
(just like TTL and 74HCT ) as expected and as I assumed. Verify
Change R10,11,12 to 5V/7mA = 720 Ω pullup to 5V for 7mA sink and expected rise time ~ 720 /3.3k *10us est. = 2.2us rise time.     
"0" Power= 5V*7mA ( 35 mW out) + 1.3 V *14mA (=18mW) * 50% duty cycle = 27mW avg vs 200 mW total rated Abs max.  ( &lt;&lt; 50% of abs. max  is good rule of thumb)
If you need less than 1us, you may consider reducing 720 Ω pullup to 5V


The PWM does have EMI filters which may be partly caused by load capacitance.  If you need a faster rise time experiment with emitter follower method instead of 720 pull up on collector try 330 Ω to emitter follower to gnd for 0 to 4V non-inv. output.

update


Get rid of opto isolators  use 50R series to match driver impedance to reduce EMI and ringing from apparent Cin load.  


p.s. use a log User Experience (UX) interface for dim control with 12 bit resolution.

p.p.s.  breadboard crosstalk is more likely 10 nH/cm for wires and 2 pF/cm  if twisted pairs and less if not.