I have good news and bad news.
The good news:
I have found that Microsoft offers a very good and inexpensive search. You can put it together and add java script to a Squarespace site using the Code Block. Works great after a couple of days of testing. Unlike Google, it's ad-free. Works great on the test.
Try it here: https://wtgpoa.org/test (password: search)
Try "park" and "fences" and "bears"
Here a link that introduces it. It seem this has been around for a couple of years.
https://www.microsoft.com/en-us/bing/apis/bing-custom-search-api
https://blogs.bing.com/Developers-Blog/2018-04/bing-custom-search-build-a-customized-site-search-experience-in-just-a-few-minutes
The bad news:
It's Microsoft.
1. It is very inexpensive, I think. I cannot tell you exactly what it cost. Microsoft's Azure pricing on it's service, but especially so on this, is very complicated. (Tip: The vast majority of Squarespace users would probably sign up for what the call "Pay as You Go.") Here's not very useful link: https://www.microsoft.com/en-us/bing/apis/pricing
2. Microsoft's UI for setting this up is a nightmare. Know a bit of coding isn't going to help you much. The toss around labels such as "Endpoint," "Keys" and "Instances." There is no "done" when you're done. There is a ton of reading they want you to do.
3. You have to set up a Microsoft account and get into Azure. Big deal? Well, good luck finding custom search when you sign in. It's all of Azure. It's one for dang thing to manage. Use the links above.
So, am I going to use it? Maybe. It's the client's call and I'll have them test it. They're going to love it and the cost will not bother them. But, I'm a set-up and leave guy. I don't offer much support afterwards. I provide them with names of Squarespace shops who will help. But, they aren't going to know this Microsoft package, so the client will be taking a risk. I'm still thinking about what I'm going to recommend.
Thought?