To not have a housewarming because of this friend

Indeed. So if the OP's friend comes out with a tactless comment like this, it isn't for the OP to feel guilt. It's up to the two grown adults involved to manage their relationship. Does it matter if the sensitive relative thinks the friend is crass/opinionated?
The only thing that occurs to me is whether the OP has told gobby friend a secret that she doesn't want blurted out at the party. In which case, OP, have a word with her beforehand.

I have a friend like this so I know just what you mean, op. Anything you tell her rebounds somehow. For example I told her dh was trying to lose weight (as was I). First time she saw him after that she said sarcastically 'I see the diet's going well then. Ha, NOT!' This was so awkward and offensive and her answer was of course, 'oh you know what I'm like, I don't mean anything by it.'

The issue is her being overly opinionated, that's fine. Just tell her she needs to be polite at the housewarming with all but XYZ of her own close mates; spell out if needed "i.e. no comments about people kids please" or whatever. If she can't manage it then she's not much of a mate really, and you'll be able to explain to her next time why she isn't on the invite list.

That had occurred to me, too. If the Op doesn't really want to have a party, she has a (somewhat thin) excuse in Sensitive Person having to be kept apart from Gobby Person.

Well, then you're suffering because you have a tactless, interfering, gobby friend.

Are you saying she behaves like this to you, though you somehow don't mind, but that if you said to her 'Look, Barbara, put a sock in it, would you? I don't care whether you think my relationship is on its last legs, my parenting is lax, and my ass looks big in this dress -- why don't you practice thinking things without saying them out loud? It's quite possible and you won't explode, you know', she would throw a tantrum?

Are you using this complete non issue—this manufactured crisis—to avoid having any kind of party? I would suggest you tackle the bigger issue which is you are taking ownership of—or have been raised to take ownership, of everyone’s experiences snd feelings but your own. In reality you are not responsible for making sure “your relative” is never upset or annoyed by your abrasive friend. Nor are you responsible for your abrasive friend not being invited to your party so she can shit sll over everyone in a pathetic bid to feel better about herself.

Try googling Mel Robins and “let then” and try teaching yourself to “drop the rope” and walk away from other people’s drive to ruin things or make everything messy.

Designate another friend to shadow her and intervene?

Have a word with both the friend and the sensitive family member?

Pick a date you know either friend or family member can’t come?

You absolutely cannot control what other people do. The only person whose actions you can control is you. You can always only have what is possible. Decide of the possible options what you want and go for that.

So your options are
Don't have a party
Have 2
Tell nasty friend to shut it and risk her turning on you
Spend the whole party worrying about the friend being a cow to someone

Which is the least worse?

Why would it be more expensive to split it down the line? You aren’t inviting more people.

Also, as someone with high levels of anxiety and ASD who struggles socially, I find smaller and more controlled gatherings to be much easier to manage.

Also; there is no rule to say you need to have any housewarming at all. When DP and I bought our house we did intend to do one but it never manifested. We hosted his parents one weekend and we did have friends over for drinks when first moved in. I couldn’t even imagine mixing the two! No thank you.

Have the party. Deal with the fallout if and when she says something. It’s not for you to audit. If she says something that puts someone’s nose out of joint then that’s between her and them.

If someone at a party had a pop at me for the way I live my life, I’m strong enough to deal with it. I might laugh it off, I might snap back, I might just say ‘totally disagree Lisa’. I wouldn’t blame the host of the party. You have no control. She might say something embarrassing but so might Auntie Brenda.

Maybe tip off the person who is a bit sensitive and say ‘Lisa is coming to the party, so you might want to give her a wide birth’. Gobby, opinionated people at parties are a worry, but they also liven up proceedings which you sometimes need.

why not?

The fact that this friend is causing you this dilemma isn’t good. I have been in your position OP. Does she ever say things that upset you?

If you were only told to tone down being constantly rude, judgemental, and offending strangers with your attitude I’d imagine you then ending the friendship over it wouldn’t be the end of their world.

So what? Why shouldn’t she be offended? It won’t kill her. You are doing her no favours by letting her destroy her social life.

Yes I agree. If you can give it then take it.

My mum has a friend like this. You just have to really tell her off for it when she crosses a line. If she’s being a rude twat tell her.
We stopped inviting her to things when she really offended somebody, for a few glorious years I didn’t see her at all. She’s reined it in (a bit) these days but I’ll be honest I avoid gatherings when I know she’ll be there.

Agree, it’s ridiculous to say it isn’t an option. Clearly OP has a friend group that includes this woman and they all know already what she’s like and are fine with it. Invite they entire group over for drinks and nibbles, however you party.

A different weekend have the whole family down for a BBQ or a meal, whatever you’d do with the extended family and any other close friends that aren’t in this group.

It very certainly is an option. This isn’t a wedding reception, it’s just a housewarming. You can definitely toast it up twice.

Since she has good qualities and you want to keep the friendship, is it really the end of the world if someone gets offended? Assuming she is not going to make racist/homophobic etc comments but just may be strident in her views I think I'd just let her get on with it. If you tell her x family member is a bit sensitive she may blurt this out so I wouldn't do that I reckon.

I understand what you are saying but if you read the thread you’d see OP’s friend can be quite nasty. I don’t see an issue with telling a friend they need to pack it in a bit becuase their attitude isn’t great to others. I personally still stand by telling a friend when they’re wrong.

I’d want a friend to tell me “X you’re being quite rude lately, you need to tone it down as others have picked up on some hurtful things you’ve said”

If you would rather end the relationship than take honest opinions on board that’s on you

I can’t stand people who can dish it out, but can’t take it.

If someone told me to 'tone it down' the friendship would be over anyway.