The reference to Kaunas is likely to the province (gubernia in Russian) rather than to the actual city of Kaunas/Kowno/Kovno.
Go to your nearest Mormon Family History Center (FHC) and get two different sets of records that the Holland America Line created for passengers departing from Rotterdam.One set of records is available on-line only at the FHC.The second set is on microfiche which likely will need to be ordered.These records contain information not found on the ship registers that were completed at the European ports of departure.
Find the church marriage register entry (not an extract) for the two sons born in Lithuania.The entry may list their place of baptism as well as a more authentic Lithuanian surname spelling.
You need to exhaust all the available U.S. records before trying to go to Lithuania for research.
The birth records in Lithuania are filed by the church where the person was baptized.There is no master index for you to search for early recorded births/marriages/deaths.