Did you get the liquid stuff from the vet or the pills from the store? If you get the pills, you can crush them between two spoons and mix in with wet food. It's also cheaper to go the pill route. I now get a BIG powdered jar from the health food store and it's even more cost efficient and less messy.
When I first got Hannah, she barely weighed 4 lbs, and was on the 250 mg, 2x day gel/paste stuff from the vet. So I would say 250mg/day would be the way to go until your little one gains a bit more weight.
Hannah is now a thriving, healthy, plump girl who weighs almost 10 lbs! She still gets lysine everyday (500mg daily) to keep her immune system boosted. If I notice her eyes weeping or if I hear her sneezing because of allergies or the weather changing seasons, I up her dose to 1000mg a day.
Of course, this has all been discussed with my vet.