Ideally, you should ask to stay WITH your puppy rather than let if be removed from your presence (esp after this situation occurred). Vets will TELL YOU it is best to remove them, so they do not associate anything UNPLEASANT that happens (like shots or blood drawing) with you....but MY dogs always feel more reassured with ME in the room with them, so I am my own advocate for them and insist they stay (unless it is for surgery or x-rays). They do LOTS better and mind me 100%, although mine have never tried to bite anybody.
2) You need a new vet. GOOD VETS do not inflict PAIN when doing shots or drawing blood, and they should always have an ASSISTANT to help hold the dog. I feel your puppy is TOO YOUNG to be TERMED "AGGRESSIVE".
Puppies at 3-4 months, have often NOT fully learned NOT TO BITE, and if frightened, the puppy may indeed become a FEAR BITER..... in a given situation (and in my opinion, suggesting the vet did something very WRONG - behind CLOSED DOORS).
3) Look up the positives and negatives of early neutering. In most VET SCHOOL opinions & that of my holistic vet, is NOT advisable to do it until the GROWTH PLATES have closed. Early neutering also causes or likens a number of other HEALTH PROBLEMS: certain types of cancer, hypothyroidism, etc. P.S. neutering is just another cheap way for a vet to "pad their BILL". Sure (unless you have a SHOW or FIELD CHAMPION quality dog) and certainly if it is a mixed breed, it will eventually do best, to be neutered (when more MATURE). But that is YOUR CHOICE and does not have to be done, EVER if you can be careful & responsible with not letting your dog ROAM or reproduce.
4) You do not say what breed or mix of breeds your puppy is. The vet MAY have an automatic BREED BIAS if it is a bully breed or bully mix. Or insisting your puppy was aggressive & that you get it neutered ASAP (aside from her obvious PROFIT MARGIN) may be her way of "covering up" her fear or ineptitude..... at drawing blood or giving shots. (C.Y.A.)
5) We have no way to know how much socialization you have or have not done w/ said puppy but puppies NEED 10 different experiences every day: sounds/noises, sights, surfaces and people (including different races & genders, with hats, no hats, in a wheel chair, on a skate board, etc. Puppy kindergarten classes are another IMPORTANT socialization feature you need to get enrolled in, ASAP.
MY Advice: No neuter until the growth plates have closed (for your breed or breed mix) get a NEW VET that comes highly recommended, and STAY WITH your puppy.... for any further exams or shots.