my ds2 had exactly this thing. he got to 15 weeks, and i was just about managing with quite frequent bf (1-2 hourly in the day) and his demands went up hugely. he fed every hour night and day. it was exhausting, everyone told me to formula feed or give solids but i have read the research so stubbornly continued, co-slept and got through it. now things are much better.
i think this is very common and why lots of people start solids so early, but research says it doesn't help. your HV is quite incorrect to advise you to give formula and rice. we got through it with just breastfeeding. i agree a baby moon would be good, might also enable you to get rest.
i tried the tea. don't think it increased my supply really, but it gave me a confidence boost, and i think alot of low supply concerns are actually lack of self-belief rather than real inadequate supply (well, was with me!).
i would also say, take your watch off and stop counting how often/when you are feeding and just feed her when hungry - even if that is 10mins after the last feed. 'routine' is an artificial concept for a little baby. that helped me relax a lot. i agree with gybegirl, giving formula or expressed milk will reduce your supply.
they start to chew hands more at this age as they explore the world and it does not always mean hunger.
7-9 feeds a day doesn't seem alot for 16weeks.
sorry for mammoth post, but your situation seemed so similar to ours 3 months ago, just wanted to explain that you can get through it! this too shall pass!