My daughter is PG and has high functioning autism (aspergers) but I know that girls and boys tend to have different quirks. It certainly sounds like he could have some sensory issues and is quirky, but it gets really complicated when you throw in giftedness and a young age. I think it would be worth having an evaluation, provided that you can find an evaluator who has experience with young 2E kids.
You could also request an evaluation through your school district, and/or other services (in particular, occupational therapy) through your medical insurance. You do not necessarily need a formal diagnosis to get helpful therapy from a private OT or your school distirct. They will evaluate and offer services based on your child's demonstrated behaviors, regardless of diagnosis. Having an official diagnosis can help you as a parent understand your child better, and it could mean getting more and better services via your medical insurance--but you don't have to wait for one.
The one thing about school districts and evaluators and sometimes even OT offices, if they're crowded, if that it can take foreeeever to get started. Autism or not, a kid with sensory issues will probably get a lot of benefit out of occupational therapy.
If you want to jump on this, here's how I'd do it:
1. Make an appointment with your pediatrician, tell the pediatrician what you're concerned about, and ask for referrals to a pediatric occupational therapist and to someone who can check on your Autism concerns (if you can find someone with experience in 2E kids, ask for a referral to them in particular). It may or may not take weeks or months to get an evaluation for either Autism or OT services, depending on your location.
2. At the same time, look up your local school district's Special Education department. You will want to find their Child Find contact, if they have one. If you can't find it, just call the special ed department and let them tell you who to talk to. You want to find out how to make a written request for an evaluation based on your concerns. (Email may or may not be ok.)
The way school evaluation timelines work, if you request this tomorrow, you miiiight get an evalation done by the end of this school year, with services starting in the fall if your child qualifies. If you wait until fall to request an eval, it's going to be an eval done by Thanksgiving if you're lucky. Which is why I suggest not waiting to make this request.