I understand that they had their own objectives, but did Germany even try to get them to launch a second, eastern invasion of Russian territory to help split the Russian army in two?
If so, why didn't the Japanese do so?
The Japanese fought against the Soviets in 1939. It was a lengthy battle which lasted from May to September.
It is referred to as the Nomonhan Incident by Japanese and the Battle of Khalkhin Gol by the Soviets.
Japanese forces were soundly defeated by the Soviets. Marshal Zhukov first employed some of the tactics that he would later use against the Germans here.
Later on the Japanese would end up signing the Soviet-Japanese Neutrality Pact in 1941. This was broken when the Soviets later invaded Manchuria in 1945.
Edit: formatting.
The Japanese army had its hands full trying to pacify China. Then the invasion of the southern Pacific took away even more resources and manpower. For Japan, striking the south Pacific sources of oil and rubber was a stratiegic necessity to ensure access to vital resources. There are no really compelling resources to get in Siberia/Eastern Russia.