Growth Accounting with variable factor shares
Interesting question. In effect, while factor shares were thought to have remained fairly stable over a long time (the first of the Kaldor's facts), more recently they have varied, particularly in the direction of a fall in the labour share.

This short paper from (2012) shows that under such scenario, a growth accounting exercise which assumes constant factor shares leads to inconsistent results. The abstract reads:


  Recent evidence shows that factor shares are not constant. Consequently, growth accounting exercises rely on a false assumption and a measurement problem arises. We propose an empirical methodology to solve the measurement issue and estimate TFP growth.


A more complete and recent analysis from the same author (plus another co-author) is here (and a theoretical model with these insights is here).

As you can see, this is a very topical issue, and there seems to be very little empirical analysis so far. I only found this one.   As the authors in the second article linked conclude, 


  any efforts to align empirical growth research with the reality of the factor share data cannot be carried out using standard techniques. New techniques need to be developed.


There might still be some time until a satisfactory empirical method is developed.