I would say that you have a very good reason to make an appointment with a Developmental Pediatrician. She has more than one of the flags for an ASD, and yes, what you describe is a very common issue with kids on the spectrum. It is very common for children to recieve a more mild diagnosis prior to being diagnosed with an ASD. Right now, SPD is a popular gateway diagnosis, just as CAPD, and OCD, and ADHD have been in the past. It is very encouraging that she has made progress, and you should not feel bad if she gets a new diagnosis, because she would have been primarily treated based on her symptoms through therapy anyway, so it is very likely that her therapy would have been the same, even if she had started with an ASD diagnosis. She is still very young, and you are right on top of things.
Call your local childrens hospital, and find a Developmental Pediatrician, who will do a full scale evaluation that will include all areas she needs, and give you a comprehensive evaluation that will miss nothing, so you will not have to wonder if you missed something. It will take many months to get in, so continue the therapy.
Next, since she is turning three, write to your school district's special education director and request that your daughter be evaluated because you suspect that she has a diasablity. That is the language that triggers their responsiblity to evaluate her. Do not depend on the public evaluation to either diagnose her, or identify all her needs. While the school may say that they are doing this, you will always want to hold the evaluation that contains her diagnosis, and know more about your child and her needs than any public agency. The school is required to provide services so that she is functional in the school setting, and you will want to maximize her potential, which the school is not required to do, so get as much as you can from public services, and supplement her program with private services so that you maximize her care up to the full program suggested by your Developmentalist.
What you describe is much like my experience with my children. While my rubbermade baby who was sensory seeking and never felt pain talked early, she scriped all day long. Our youngest was nonverbal until she had intensive speech therapy, and she had much difficulty (still does) cominig up with the word for the item or thing she wanted, it is a categorization, reteival issue that is common for children on the spectrum and many have a great deal of trouble with nouns and use indefinet pronouns for everything, which makes speech very difficult. Other oddities that you might look for are using the "ing" form of a verb to identify objects (ie, the "cooking" for the stove or oven) and the misuse of pronouns.
Good luck to you, I would do make thise contacts this morning.
M.
PS: www.wrightslaw.com is the best resource you will have to understand special education advocacy in the public school. Start reading both about ASD and advocacy and always try to stay in a good working relationship with your school district while you advocate to get as much as you possibly can.