I can also see exactly on bus a day online.
There are more than that.
Have a look at this link from euroticket.pl
This gives the times from Suwałki to Kaunas on the E - 614 Poland - Estonia route.
Within this link there are also times given from another company, ecolines,
I can understand why the 1.50 am to 5.00 am bus might not be too appealing, but there are daytime ones too.
Just pick a day when there are more options. I can't think how else you can get there unless you want to get a taxi, which will be much more expensive.
The times and destination I entered haven't shown up in the links for some reason, so just enter the details.
On some days there are 3 buses available.