Shoot Vehicles(KV-8/IS2 etc) with it, individual infantry cost so little that they don't provide Sturmtiger enough exp to vet. Same reason Stuka vets up slowly unless you manage to hit vehicle or destroy team weapons with it. Use Panzerfusiler AT nade or Fallshcrimjadger panzerfaust to slow down enemy vehicles will help you hit vehicles otherwise hard to hit(like T34).
i use fortification doctrine as getting ST will cost me a panther so i build a pak and get some raketns to cover its retreat. i did try placing Strummis mines to disable vehicles and i killed a buldozer sherman in one shot. dint notice how much it vetted it up. ill try vehicle sniping from now on.. hope my ally doesnt go mad as i will lack AT.