In Japanese, some6 initial consonants become voiced when they occur internal to some5 compound words, e.g:
This process is called rendaku, and the conditions under which it applies are complicated, not being entirely predictable.
This topic is further complicated in that rendaku can be dialect dependent, and its prevalence has evolved over time. As such, early 20th century romanisations of technical martial arts terms may not reflect the common pronunciation in many contemporary Japanese dialects.
This survey by the NHK Broadcasting Culture Research Institute specifically notes a compound word involving 払 as being 'unstable' in pronunciation, with 68% of those surveyed now pronouncing this word as barai:
The target words have been chosen from vocabulary that are unstable on word-form and controversial when they are pronounced on broadcasting, including voiced-form and unvoiced-form on Rendaku.
未払いc mi -Harai(30) / -Barai(68) 【c: 1,307 persons, 2007 Mar】
As this is only a poster presentation, I don't have access to the complete data, but it implies that the difference is age-specific:
In this presentation, the item which shows the age difference will be taken up.
Since multiple Japanese dictionary sites1 2 3 list both pronunciations4 in their entries for 出足払, it appears that either is acceptable (with the possibility that barai is more common among younger generations).
5. E.g. kata-ashi-dori, but obi-tori-gaeshi