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 particularly. 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 development of Russian.
Of course, that does not forbid the possibility of some individual having, in some particular situation, poorly worded some particular french phrase, but that indeed demonstrates that this fact would be nothing but anecdotal.
Of course, if the Russian elite can and actually pay for authentic (and sometimes renown) french Professors, less fortunate classes of the society will resort to classroom sessions sometimes held by all sorts of people including deserters, adventurers, real-false-churchmen... well... people whose relationship with French was everything but academic... This could well explain the possibility for some Russians to have learnt somehow approximative grammars or to speak in a colorful non totally academic style.