Your wife unfortunately appears a little sexist. Because it is a female, your wife trusts her and "assumes the best" about her intentions.
But if it was a man kissing your daughter on the lips on camera, your wife would be howling, sweating, screeching, and reaching for the nearest pitchfork and dialing 911 at the same time. (And we all know it, too.)
In reality, over 50% of all domestic violence occurs by women, and a majority of child abuse cases occur by women. Women are not "less likely" to harm a child- at least, not significantly statistically. It is very close, in terms of gender instances of sexual or physical violations to children. See all the teachers molesting or having sex with elementary or middle school boys? We cannot pretend this is only a "male epidemic" here. Your wife ignoring the obvious reality about the fact both genders can and will harm children, is putting your child at risk.
Your wife is being like the racist who says "Oh, uncle Johnny touching him in a weird way is okay- after all, he's our race, our blood, so we can trust him! Relax!" Then next thing you know, the kid is in therapy or suicidal because he was molested and has real issues. You dont want that.
You are not wrong to feel the way you do- You feel the way you SHOULD feel. Her kissing your daughter on the lips is inappropriate, borderline pedophilia, considering they are not family or blood related in any way. Child care can involve hugs and cheek kisses, but kissing on the lips crosses the line. It also could expose the child to a disease/STD. What if the woman did not know she had herpes? What if it was dormant but passed to the child? Kissing on the lips, regardless of what she claims her intentions were, crosses the line.
And your wife ignoring this based on sexist bigotry, is putting your child at risk.
I recommend you first show your wife my post, explain that you are going to talk to the babysitter, and ask that you hope you have your wife's agreement on this. Explain to your wife that even if you dont, you have to stand up to protect your child whether or not your wife is willing to. That you are the man, and you are going to do what a man needs to do- protect his family.
After you have explained this to your wife, then you should go talk to your babysitter and tell her to never do this again, and tell her you dont care what your wife says, that if you see it again you will contact the authorities so it better not occur. Tell her she's free to keep watching your kid but you dont want any more lip kissing, regardless of if it's innocent or not.
And if the lady quits because she cant kiss your daughter on the lips anymore, then well, that tells you all you need to know about her intentions.
NOTE: I am not saying that you need to blow this way out of proportion. Obviously, you should be respectful, calm, and honest when talking to the babysitter. Explain how you feel. but also explain that you understand its possible the babysitter did not mean any harm by it. Ask the baby sitter to explain herself and her culture. If her answer is satisfactory, then you can tell her she can continue babysitting- IF she makes sure that this thing never happens again. And if she doesnt give a satisfactory answer, or starts getting crazy or defensive or acts super weird about it, then maybe you should tell her you respect her but you're going to need to find a new babysitter.
I don't recommend "filing charges"- that would be over the top, of course, on something that you really cant prove she had any bad intentions on.
However, "what you can prove in court" is a much higher burden of proof than the "something being sketchy enough that you dont want them around your 3 year old" burden is. With your 3 year old, if something is even slightly off, you probably should play the safe side, as long as you arent harming someone else. In this case, finding a new babysitter doesnt really harm anyone. The baby sitter can always get a new job. but you cant get a new child.
Everybody can handle this peacefully and respectfully- nobody needs to get in trouble- no hard feelings- but if, after talking to the babysitter, you still feel she isnt a good fit for your child, then you probably should find a new babysitter. You dont have to explain yourself to others, except perhaps your wife- because your family comes first.