I probably would go Buttler, but it's not an easy choice at all. In absolute terms I haven't been particularly impressed, but in relative terms I think only Woakes (yes, I know) averages more than him since he's come into the side. He is, by the very low standards England have set over the past year, pretty secure. Only Stokes is probably more so.
The problem for Burns, for me, is that he averages under 30 still and two tests ago people would have dropped him. I like him and I like how he approaches the game opening the batting, but he needs to get his feet under the table first.
Buttler might not be the long term answer, but if we get two years out of him where he oversees a change of culture for the side (a la Paine, but more about remembering how to bat rather than how not to be cnuts) then hopefully the picture sans Bayliss will have settled down a bit.