A decade or so ago I'd have agreed with you here. But today this isn't really the case.
For all the bragging rights they would merrily fight each other over a trade/defence deal. I remember the MMRCA-1 in its final stages, the then French President Hollande remarked how UK doesn't have a much of a defence industry and thus India should buy Rafales. To which the then British PM Cameron hit back saying Brits never ran away from fighting unlike the French or something like that, I don't remember exactly. The point is, here you have the leaders of two permanent members of UNSC fighting over who gets to sell a few planes to a third world country.
More recently we had the US and Russians going at each other over who gets to sell AD missile systems to India. You know what I take home from all this ? Money talks. Every time to everybody.
As a famous movie once taught us :
" Baap bara na bhaiya
Sabse bara Rupaiya "