I've done it both ways many times and I'm not sure the risks are greater with one method rather than the other! A router can kick a little when starting, or wobble on the plunge, either of which will compromise the cut. But equally you can slip with a scratch stock, and it's also much more prone to tearing out the cross grain veneer.
Personally I prefer the router, especially for the specific application you mention as whatever cutter you use it will be engaging into two dissimilar materials, veneer on one side and lipping on the other, in my experience this really does make a scratch stock difficult to control and (at a micro level) it'll be bucking and skewing in your hands even after you get the cut started. See my later link to the Garrett Hack style scratch stocks which go a long way to overcoming this problem.
Having said that it also depends on how big the box is, as I'd favour the scratch stock for smaller workpieces where the router may be less well supported. If you do use a router it's important to make sure there's good support throughout the cut, which may mean building a run-on/run-off frame exactly the same height as your box, to which you can attach stop blocks to position the router precisely.
And to really muddy the water, the choice isn't just between scratch stocks and routers, there's also other hand tools like this,
https://www.lie-nielsen.com/product/too ... ?node=4180
or like this,
http://www.leevalley.com/US/Wood/page.a ... 43314&ap=1
or even like this!
http://www.leevalley.com/US/wood/page.a ... at=51&ap=1
I've used the Holtey purfling cutter and the Lie Nielsen inlay cutter and they both have the advantage of
scribing the edge and are therefore that bit cleaner across the grain. But by carefully going over the workpiece with a cutting gauge you can also scribe lines, and use the scratch stock to remove the waste between the gauged lines.
Finally, most British scratch stocks have a pretty short fence. They still have their uses but I'd encourage you to look at this alternative, Garret Hack style of scratch stock.
http://www.finewoodworking.com/how-to/v ... stock.aspx
I've been using one of these for about a year now and for straight line cuts it has two huge advantages, most obviously is the longer fence, but also the way you can rock it fore and aft to present the scratch stock's blade to the best advantage, that can make a big difference.
Good luck!