I am in no position to give any advice as I am more or less a novice, but I did want to let you now that I had a very similar situation that could have ended badly if my daughter and son-in-law were not such responsible and knowledgeable dog people. My dog, a timid, laid-back, gentle corgi, lunged, teeth-bared, and growling at my daughter's dog, a large, very goofy but very sweet, yellow lab. The lab was being friendly and happy but "in her face" in no way threatening. I immediately said "No" and quickly but calmly put Sully, my dog, in the bathroom for a couple of minutes. Then I took her out, but put her back (calmly) every time she growled. This all happened just after I adopted Sully and when she was meeting family for the first time. What helped enormously was neither dog owner being defensive and the fact that the more aggressive dog, Sully was also the smaller dog. At my daughter's suggestion we took both dogs outside to a more neutral area and let them "work it out" without the dynamics of protecting their territory and special people to add to the tension. I have NO idea if that will help in your case, but I remember being very upset and afraid all future family gatherings were doomed. Once Sully truly realized the humans are in charge she really worked to control her resource guarding, but she still tries to growl when another animal is getting attention from any of her favorite people. I had to get guidance from an animal behavioral professor and student at our local college but that minimal assistance and my family's calm and confident manner got us through that initial rough patch that could have become a huge unresolved issue if calmer heads hadn't prevailed. I my case, I was the least calm as my sweet little dog took on demon behaviors, but she was reacting out of fear and a natural instinct. I just had to let her know we are in charge so she could relax. I am betting the "Nothing in Life is Free" philosophy would also be helpful. I have heard good things about it and it mirrors many of the things I used to help Sully learn to trust me as her "leader." Again, I am not an expert, but this situation, if it is similar to what I went through, is very upsetting to witness, but much more manageable and less serious than it may seem. I hope the same is true in your case. I did keep my dog restrained by the way until I was sure she wasn't going to actually attack. Again, I sought behavioral advice and I had the blessing of the other pet owners before I made any attempts to get the two dogs together, but it was much less traumatic than I expected. I truly hope you have the same luck!
Oh yes, been there before. When Wynn was younger and I would take him into this large pet store that usually had other dogs there Wynn would walk up to and growl at ONLY the BIG dogs...I finally figured out he was "feeling" my stress/discomfort after this happened a few times. After this I realized that I had to talk myself into being calm and less stressed. I don't know how/why but I believe Wynn was reacting to my reactions. I changed and he did too:) Never a problem again. Strange but true.
Not so weird. They figure out immediately which dogs are likely to be a threat to attention and a special place with the people and places they consider their own special loved ones. Sully has learned she has to tolerate my daughter's dog, Remy, and she is never aggressive toward her even though Remy is much larger. She isn't afraid of Remy retaliating, but she is worried that I won't like it. As long as Remy isn't trying to get my attention or Remy isn't getting attention from any of the people Sully loves, there is no problem. If my daughter or I are patting either dog, the other immediately tries to worm her way in for some attention. The cats are never a problem since I'm not really a cat person so the dogs ignore them as I do. If a dog is getting attention from a person Sully doesn't really like she has no problem at all. Two of my granddaughters are identical twins, but Sully is only close to one. She totally ignores one girl, but she is much more possessive of "her" Cloey. It definitely isn't Sully's most attractive quality! We are always working on it.