If you think it would be "in character" for you, then just order some "23andme" kits, and don't even ask for opinions. Just say you're interested in what you can learn about yourselves and the others. It's not as reliable as a paid-for DNA test from a solid, reputable lab, but it's still pretty reliable. I mean, if your son doesn't share any DNA with you, the only way something like 23andme could get it wrong is if they used the wrong samples or got the results mixed up; the test itself that they have will absolutely show no relationship if there is none. If the results come back and they show anything other than a very high percentage chance that you're the father, you could then navigate the waters of asking for a more professional test.
It depends on your family dynamics, and your personality. For many people, doing 23andme would be completely in character and not raise any alarm bells. For other people, it would seem really strange, and your family might wonder if there's an ulterior motive.
A variant of this would be possible if you have a near relative (brother/sister/parent) who you trust enough to share your concerns with. If so, that person could pretend to have gotten really interested in genealogy, so much so that they decided to pay for a bunch of 23andme kits, and are asking a bunch of relatives to participate. They could say that they're starting with themselves and your family, and if the results are interesting, they'll move on to other close relatives. Then you'd just pay for your brother/sister/parents tests, as well as the tests for you own family. If that's more plausible, and you're not worried about the relative you take into confidence breaking your trust, it might be away to get around them suspecting you if it wouldn't be in character for you.
Depending on the relationship you have with your son, perhaps you could talk to him and keep it between the two of you. You could say that your feelings for him will not change at all based on DNA results, but if you're not his biological father, it's important for him to know, so that he can learn about any medical issues or genetic diseases that run through his biological father's side of the family. Obviously, how this would be received would depend hugely on the relationship you have with your son. I can imagine some sons who would completely understand your concern, and would understand that you think there's only a very small chance of you not being the biological father, and would understand that saying anything to his mother would cause a lot of hurt and damage for no good reason. I could also imagine sons who would want (or feel obligated) to tell their mother that you have these doubts.
I don't know much about facial matching, but I would be shocked if it's got anything greater than like a 80% chance of correctly showing that two people are unrelated. I'm pulling that number out of my ass; I just think if it was that easy for an AI to do it to a high degree of confidence, that human beings would be able to do it with a pretty decent amount of success, and the world would be a very different place, given the number of children that are born and raised by husbands who don't realize they aren't the biological father. I've seen estimates of up to 5% of children being born this way, which I don't really believe. But I would find 1% plausible, and that's a *lot* of kids; I really think if facial resemblance was enough to go by, we'd have seen many more fathers finding out than we historically have.
So, if it were me, I'd try to get the testing done without arousing suspicion. It's hiding the truth, but in this case, I think that's the better option. If I couldn't find a way to get it done without making my wife suspicious, then I'd have to think hard about whether to tell her about it in order to get the testing done, or else just let it go, and enjoy my golden years without potentially stirring up trouble. For me, I'm insecure enough that I'd probably need to know, even though the rational part of my brain might be telling me that I'd be far better off just letting it lie; the uncertainty would wear me down. But I think if I had better self-esteem, I'd be able to just let it go and enjoy my marriage and my son without investigating.
Whatever you decide, please update us. This sort of thing is going to get more and more common as time goes by, and it would be really useful for regular members here to learn from some real-life experiences, so that we're better able to advise others in the future.
I hope this works out for the best.