SET vs Push-Pull (Line Magnetic vs Primaluna)

I can't answer your question in regard to LM amp. I can only say I ran Coincident Turbo 845 SET with Merlin VSM-MM, a similar load to your speakers, quite satisfactorily for years. I doubt any other SET tube type would provide enough power.

I  have a Prima Luna Dialogue Four modded with Takman resistors and a variety of boutique caps. I also have custom built 300B SET monoblocks and the Coincident 845 SET which all rotate through system. The largest difference between the SET's and PL push pull is the sense of immediacy, a greater illusion of performers in my listening room with the SET's. Whenever I listen to push pull tube or solid state setups I always get a sense of being more distant from performers, I'm taken to the recording venue rather that the recording venue being brought to me as with SET. My mind interprets these systems as high fidelity. I always hear live performers in room with my system, just slightly less with the PL (this all with my highly modded Klipschorns which excel here). I've not listened intently with all three amps with my Merlin VSM-MM, a speaker more similar to yours. Based on memory I recall hearing this same jump in immediacy when I went from Cary SLM 100 monoblocks, Conrad Johnson MV-55 and Nirvana Electronic Works A-60 (battery powered class A solid state) to Coincident 845SET with Merlins.

If you were looking to get closer to SET sound, move to KT will be move in somewhat opposite direction, imo. I prefer the  EL34 house sound to KT 88 in my PL Dialogue Four. KT presents as more precise sound, less euphonic. PL amps are in modern sound class of PP, KT put it over the edge in my system. With change to KT I'd expect you'll get better bass control and extension, greater treble extension, but you'll lose a bit of the EL34 mid range magic.
I don't know which EL34 you're using, but the best I've heard are 1990-early 2000's Svetlana and/or SED. I presently use  SED Black Sable (early 2000's, cryo'd and closely matched limited production), really not that far off SET sound with these tubes. Present production Svetlana is not the same tube as these, ones I'm talking about are now sold as vintage.
Still, do try the KT, although I'd suggest the KT77, seems many prefer the lower powered KT's as best sounding.