Ok I got that wrong. So Barney and Pippi are used to each other, all good.Pippi and Barney are mine. Keb is my mums.
So Keb is the visting dog and is much the same size as Barney ie potentially a good play match.Barney is as best we can tell a lab x. Pippi is a foxy/shih tzu mix. Keb is a NZ Huntaway/Beagle mix.
And Pippi is fear aggressive - I agree with your reasoning. And it is common in little dogs - big dogs can be scary and dangerous.
And Keb gets a bit protective around his owners. Argh. It would be better if he was comfortable around the other dogs and didn't feel the need to be protective or jealous (of attention given other dogs by his owners).
Disclaimer - I am not a professional dog trainer - I don't guarantee these suggestions will work in any way:
Loads of tiny pieces of roast chicken (or other yummy soft treats) in exchange for good behaviour over a series of training sessions might help. Again I'd work with one dog at a time. Not all three at once.
So a possible session might go - with Barney away (out of sight and hearing) and with plenty of room. With Keb and Pippi both on lead.
Have Keb and owner(s) sit at one end of the yard or neutral space, and have owners feed Keb treats for being nice (not for growling).
Have Pippi at the other end with her owner and feed her treats for being nice. If she can be nice at a distance, approach a little bit, with you (the owner) between Pippi and Keb, and feed her more treats if she comes to you and stays nice. Continue to approach in this way. Stop if either dog becomes the least bit excited in any way about the other dog. Stop way before it becomes barking and growling, stop when either dog looks at the other dog and can't easily be distracted by the chicken. Reward attention on you, reward less for attention on the chicken. If one or both dogs do not calm down, go back to a bigger distance.
Work on that in five minute sessions and have a (long) break between. If you can fit two or three sessions in each time Keb comes to visit, that should help Pippi. And you can work Pippi in the same way with dogs out and about - just maintain enough distance that she's not getting stressed.
And try to keep the lead loose, if it goes tight, you're too close and Pippi will feel her options for flight are restricted, even if she's pulling forward aggressively - so she will go into fight mode.
So I'd work Keb in the same way as Pippi. You need one dog as target - in this case you could put Pippi away and sit with Barney - while Keb and owner tries to approach you and Barney, and both dogs get treats for being calm and paying attention to their owners. And you only close the distance when both dogs are calm.
Depending on how the dogs are, you may need a bigger space than a back yard to work this.
I find a lot of fear aggressive dogs will do a polite greeting with another dog that holds a completely non threatening pose (eg upside down with paws in the air or at least a steady drop/down position and looking away) as long as the fear aggressive dog is the one allowed to approach in her own time on a loose lead. If she doesn't want to approach - don't force the issue, just wait. And you need to start from far enough apart that the little dog is not stressed. You might need several sessions over days to get this far.
You may also want to check out Turid Rugass on the net, she's very good at explaining doggy body language and polite greeting techniques eg curving approaches, much looking away and sniffing. This can be faked by putting treats on the ground and walking the dog on lead in a curved line. And if you can get the "target" dog to drop or roll over into a non threatening position - so much the better for the anxious dog.
Turid Rugaas - Calming Signals Community