Heyo. Sounds like you're in the sort of hell I find myself in quite often.
If I'm not mistaken, the 5879 is somewhat similar to the EF86. I just built an amp with an EF86 in V1, and in my experience, they don't stay clean for too long. Granted, I could be doing something wrong, but they don't don't tolerate a high signal input as well as a 12AX7. The break up in mine is quite pleasant, so I don't mind it, but I have resigned myself to getting considerably less headroom with a pentode in V1.
I found a sorta fix by running global NFB from the speaker output to the cathode of the EF86. That helped a lot (used a very large resistor for minimal NFB - 680k), but only because I also made the 22uF bypass cap switchable. With the bypass cap in, I get max overdrive - kinda cool. With the cap out and the NFB in, I get more headroom and a tamer tone. It is an on-off-on switch, so in the off position, I just get the 2.2k cathode resistor.
Now, I claim no expertise, especially in this crowd who are so highly knowledgeable, but I have learned to lower my self-imposed expectations and experiment to see what I can learn, rather than what I can make. The above results gave me a blast. It's not the amp I was hoping for, but many guitarists would find it - at the very least - interesting, and quite playable.
Hope that is some help. Oh, and speakers do have different break-up levels. I like old alnico speakers, but I believe the magnets can go flat, resulting in early breakup and a lesser tone.