You really need to have four layers so you can have either two ground planes or a ground plane and a power plane. This is required to keep the impedance on the signal lines somewhat constant. The reason I say "somewhat" is because it is dependent on the trace width and the layer spacing. You are filling the top and bottom layers with copper for a ground plane and that will help but it is not a good as a proper plane below these planes. If you have a look at the data lines coming into the part. The interior data lines, 2-7, will not have a ground plane anywhere close to the signal trace. So you may end up having some clicks, pops and other digital "glitches" due to signal integrity problems. For the other signals it will be difficult to predict what the signal return paths will be without a ground or power plane.
The clock and data connectors should have ground on every other pin so the signal lines have ground on both sides. I think you are doing that on the SDATA connector but it is difficult to see if that is true. You should also do the same with the clock connector.
I also see a problem with C25. You have vias between the cap and the pins of the part. You need to change that so that the cap is the first thing the trace hits when it leaves the pins. Then after the cap you can then place the via there. This ensures that the cap is the most effective at keeping the noise out of the part and keeping any noise generated inside the part from causing issues elsewhere in your circuit. There may be other places that this happens. I see a few small vias near the pins so anywhere you need to filter the pins, the CM lines, the FILT pins or power pins, you need to make sure there are no vias between the filter caps and the pins. Sometimes you may not have a choice but usually I have found ways to do it. Having power and ground planes helps to give you options.
So when you are ready to build the production version I suggest you go to four layers. I understand the need to keep prototyping costs down!