So the growth isn't painful if he's eating with no issues, that's good at least...I'm just trying to figure out this "code" your vet gave you, lol...the fact that she has "a growth" is pretty obvious, that one I get, lol, but they have absolutely no idea what the growth is if they did no needle aspiration/biopsy. So it could just be a cyst, it could be tumor, or it could be an abscess. So they basically have no idea what it is and don't seem to really care either...the Pseudomonas I'm ignoring, as #1 Pseudomonas is everywhere and is very common to find in a mouth and everywhere else, and it's an "opportunistic infection/bacteria", meaning that unless she has an underlying disease or condition that is compromising her immune system, the Pseudomonas will cause her absolutely no issues, and not cause an active infection at all. So forget about it...now they said she has "a secondary infection that is incurable"...you need to ask them what kind of infection this "secondary, incurable infection" is. That's what you need an answer to. And you should also ask them where this secondary infection is: is it in her mouth, in her throat, in the growth (abscess), etc.? So I would start there, because that's going to solve a lot of this. What is the "secondary, incurable infection", what bacteria or fungi or protozoa showed up on the culture, and why are they saying it's "incurable", and where was this particular infection found?
Honestly, I think they are bullshitting you, I think they only found Pseudomonas on the culture (because they failed to do a needle aspiration to actually diagnose what the growth is) and they know damn well that Pseudomonas is not causing her any issues, so they basically don't have a clue what is wrong with her but are afraid to not do anything or to not put her on some kind of antibiotic, and they're thinking "she's already 9 years old", so I think the vet just said "tell her she has a Pseudomonas infection, which is true and not a lie, and tell her that she's got some other, secondary infection that cannot be cured, and we'll just prescribe a broad-spectrum antibiotic and be done with it".
If you call them and ask what microbe is causing the "secondary infection" and ask why it's incurable and why she needs to be on antibiotics for the rest of her life, and they cannot give you a direct answer, like "The secondary infection came back as Staph Aureus and it's in her mouth" (just an example of what they should be able to tell you if they really did in fact diagnose another infection in her mouth), then you'll know they are lying and have no clue what's wrong with her.