Today, he's having behavior issues at school. I think a lot of it is sensory related (currently fighting the school for an SPD evaluation, I believe he is sensory seeking) BUT I am also wondering if some of it is boredom. As I began researching trying to figure our what his issue is I see that some gifted children do have some behavior issues.
I didn't want your post hanging out there with no response. Believe me - we have some amazing experts here who can give you perspectives on how testing can help and how to advocate for your child at school.
But first - behavior issues. I want you to have some comfort here. I don't think anybody has ever done a poll here, but I'm betting that a HUGE percentage of highly/profoundly gifted kids have behavior struggles in elementary school and beyond. My DS8 (who is in second grade and is a Davidson Young Scholar) sure does. And if I gave you details, I'll bet you'd be raising an eyebrow or two!
There are lots of factors making it tougher on these kiddos. First - you have a boy, and elementary school is just generally stinko for little boys. And he's a very bright boy, bright enough that he feels different, probably. And that's hard, too. Especially when he gets peckish

And - if you read much about giftedness, you'll learn about asynchrony. These kids develop on paths that are NOT lock step with what you are told to expect. In some ways they develop faster than typical. Some slower. It only serves to give you and them a bit of whiplash when trying to navigate this new world called "school."
And - HG/PG kids are often extremely sensitive. Read up on Dabrowski's overexcitabilities to learn more about this. You're already headed in the right direction, given that you're checking for sensory processing issues. This also plays out in extremely strong feeling about many things. There is a beautiful essay that captures this - beginning with the words "A touch is a blow." Maybe somebody knows the article and can link it.
Finally, many HG/PG kids come bundled with a learning disability. They are referred to as twice exceptional (2e). That adds an extra level of challenge. E.g., my son is dyslexic (even though he reads grade levels ahead so the school struggles to understand it) and dysgraphic (and everybody who sees his handwriting understands that RIGHT AWAY!!).
This can all make raising a HG/PG kid challenging and somewhat exhausting. On the other hand, the HG/PG kids I've come to know are a joy and a bit of a miracle to see, in my opinion. It's stunning and exhilarating and inspiring to watch them experiment and learn. It's a wild ride - but one I'd never want to miss!
I could go on and on - but look for the "you know you're raising a gifted kid" thread and the "quirky" thread on this board. I think you'll find some of the posts resonate with what's going on in your house.
As to testing - it's hard to know when/how to invest in this. Others will have more educated advice here. I'll just make a couple points. First - when you decide to do it, go to somebody who is experienced with gifted kids. It will be worth it. After we did, I realized we were throwing away money when we talked to specialists that didn't understand HG/PG. Second - for us, we finally did a full neuropsychological assessment. I'm not exaggerating when I say it changed everything for us. You can search on my posts, and find details. But bottom line - super expensive, and absolutely, positively helpful and worth it.
I totally banged this out while DS was in the tub. Sorry if it's incoherent. Just wanted to reach out and say "Hi, Welcome, and you're in the right place!"
Sue