You can't do that with BTC -> ShapeShift XMR -> XMR.to unless one of those services were taking a 3.78 BTC fee (not recommended to use such as service). You would need to use your own wallet in between.
Also, I'm not sure what sort of mix factors and such being used by shapeshift, so the normal caveats about Monero being optionally anonymous (and not fully mature) apply. Again putting your own wallet in the middle and being careful with timing could give a better result.
If only there was an untraceable, unlinkable private currency which you could keep that wallet in.
Edited above
It's a little scary this precipice we're now sitting on. Eventually baby birds have little choice but to risk great pain... I am sure they never feel quite ready.