To me, this is the clue:
"He has stated that the two of them would like to have
dinner with my family, my mother, and my mother's
husband. He has asked me not to tell my mother, as
the mutual friend wants to surprise her."
On the assumption (from your question) that the friend doesn't have empathy or relational issues, this seems to be key.
Your father has either by chance (innocently) learned an old contact wants to meet your mother, or has deliberately (calculatedly) decided to suggest and set up such a meeting. So my initial question is, has your father set this up or encouraged/supported it, to provide a justification/opportunity to meet with your mother, or to hang it onto another matter, or to "sweeten" it? That's hard to word well, so I'll put it a different way:
Did your father, understanding that this contact would enjoy meeting your mother, mistakenly think "well, I can visit my son's family and make good use of the journey, and I'm sure his mother will want to see me as well if we're both there".
Or did your father think, "I want to meet her, and I'm sure she will want to see me, but if someone else is there who she likes, this will guarantee she will agree if she had any doubts (or will ensure she doesn't decline to go along with it)".
In either case, also ask yourself, did the suggestion that it's a "surprise" actually originate from the friend, as your father has implied, or was it actually suggested by your father in the first place?
Was it suggested because "this would be nice" (innocent good nature) or because it deprives her of foreknowledge and choice, and ensured what he wanted would happen (calculated manipulation)?
(Surprise visits aren't always for bad reasons: about a month ago I looked up a friend I haven't seen for almost 20 years and felt bad about dropping contact, and did it by surprise visit rather than email or phoning in advance to maximise their smile. It did! But in these circumstances I would see this as a concern)
Short version answer:
You absolutely don't need to be in the middle and should avoid allowing yourself to be manouvered into that position.
It will stress you, get backlash all round (including your wife/onlooking kids), poison things even more in future, upset everyone, and get you blamed and seen as less trustworthy/complicit against them, by everyone else - by your dad because it didn't work, by your mother by betraying her by keeping it secret when you knew she would hate it and want to know, and would have avoided if known, possibly even by your wife/son depending how they see it.
Bearing this in mind, my thought is that you can easily set up a lunch with your father and the contact, but also tell him it cannot be a family lunch with your mother and to arrange meeting her separately, because it went so badly last time (or some other pretext). Tell him you will only have an event/party where both are invited if it's a family event like a wedding or other major event, or if both of them have told you it's okay. Tell him that she has said she doesn't like surprises (which she has probably implied if not said explicitly). Say that if he wants a private meal to which she comes, you are happy to (if your wife is agreeable) but not as a surprise; you will want her to confirm its okay herself first. That's a decision and not going to change.
If you have direct contact with the contact, perhaps it would be worth quietly briefing them ("It won't go down well due to bad feeling and I don't think my father realises this, I don't want you to be embarrassed or put in a difficult/embarrassing position"). As they seem to be a friend of both parents and (from contact with mother) also empathic, they may get this and also understand why its sensitive and to keep it to themselves as a heads-up.
If you can somehow contact them, that might well be best all round.
Your wife and son:
Last, if you cant do anything else and feel under pressure to invite any of these people to your home at the same time, discuss the situation with your wife first and only do it if she willingly agrees (and if teen or older, and your wife agrees it's appropriate, your son agrees after that). This above all.
Your relationship with your mother and father is well defined and probably won't impact immensely however things go, although there may be stress, anger, blame, or upset. But you absolutely do not want to do anything that drags your wife, son, or home into it by having the visit at your place if it goes ahead, without agreeing with your wife, because this involves her (its her home and 'safe place', and her in-laws visiting, and possibly anger if/when your mother finds out she's been tricked).
She is surely aware of the position, and knows them well (I'm sure you and she have spoken at length about your parents).
Them being out if the house might also be an acceptable alternative - perhaps your wife or son would be fine with both parents having a meal at home if they don't have to be there at the time. This is separate from your mothers consent and the fundamental concerns, but might be worth noting if for some reason both of your parents do agree to a family meal but your wife or son feels uncomfortable.
Whatever you do, ensure its with your mother and your wife's free agreement and if your wife says no, accept it. A good way to do this is to tell her explicitly that if she says no, it WILL be no. That's the relationship you need to protect and avoid harm to, above any other, in this situation.