Flyback converter can't power load
If your schematic is correct then you are using the DA2032 part. From the datasheet it appears to be a 3A part.
Taking a look at the datasheet of the LT3757: they recommend taking 20% of the specified current sense (100 mV minimum).
On top of that the RC filter at the sense input lowers this sense voltage more.
Based on this: you should work with 20% of CS min (100 mV)
And then using the equation specified on page 14:

Cs_actual= (100 mV*0.8) - (65 uA*22) = 0.07857

Ipeak max transformer = 3A
So, rcs should be close to 0.03 instead of 0.02.

My first guess which is supported by the analysis above is that you were saturating the transformer. You also mention that you tried lowering the rcs but it seems that it should increase.

EDIT: Based on your findings, I took a look at your layout. There a few things that are not recommended and could be the reason of the issues your are facing.

The current sense once again : 


The location of the capacitor makes the RC filter not effective. This capacitor should be close to the IC and it should ideally be connected directly to ground pin of the IC (instead of the loop it is making now via the traces and ground plane ).
From CH2 you can that at the moment you switch on Q2, there is a high peak current which can cause malfunction and issue for the IC. It is important to filter this high peak current out. 



HIGH DI/DT LOOP during on time of the FET
As shown in the picture above, the high di/dt loop goes right through/below your IC.
In general IC are sensitive to these high frequency currents.
These high frequency current loops should be as small as possible and never below/through the IC.
That should be the focus during the layout as they can infiltrate into the IC among other issues.