My take of situation is this. Shinkansen (bullet train) does not serve directly to Mt. Fuji and Hakone. Closest Shinkansen station is Odawara, east of Hakone. Mt. Fuji is located northwest of Hakone.
If plan to take Shinkansen both ways then after visiting Hakone and Mt. Fuji have to track back same route all the way back to Odawara, which I do think is not efficient way.
It does make sense that from Tokyo take Chuuo Expressway by bus and head directly to Mt. Fuji. Then visit Hakone and come out at Odawara to catch Shinkansen.
However, if you want to do Hakone only, no Mt. Fuji, then it become more probable to do Shinkansen both way.
Sounds like you are looking to take a tour, not travel independently. If you look for a tour which visit Hakone only, no Mt. Fuji, then you may find a tour which take Shinkansen both way.