How much is known about the French spoken at the Russian court? Did French elites leave any record of snickering at the language of the Tsar's diplomats?
A lot is known and well documented from both russian & french sources.
I cannot tell anything regarding snickering particularily. As Minaev suggests, russian elites were speaking french (and german) perfectly.
According to Puskin & al., «Toutes nos connaissances, toutes nos notions sont puisées depuis notre enfance dans des livres étrangers, nous sommes habitués à penser dans une langue étrangère, plus précisément en français.»
Still according to him, «l’usage généralisé du français et le mépris du russe» is the reason grounding the late developpement of russian.
Of course, that does not forbid the possibility of some individual having, in some particulat situation, poorly worded some particular french phrase, but that indeed demonstrates that this fact would be nothing but anedcdotical.
Of course, if the russian elite can and actully pay for authentic (and sometimes renown) french Professors, less fortunate classes of the society will resort to classrom sessions sometimes held by all sorts of people including deserters, adventurers, real-false-churchmen... well... people whose relationship with french was everything but universitary... This could well explain the possibility for some Russians to have learnt somehow approximative grammars or to speak in a colorfull non totally academic style.